vr\ 


T2. 


NAVIGATION 


THE  MACMILLAN  COMPANY 

NEW  YORK   •    BOSTON   •    CHICAGO   •    DALLAS 
ATLANTA   •    SAN    FRANCISCO 

MACMILLAN  &  CO.,  Limited 

LONDON   •    BOMBAY   •    CALCUTTA 
MELBOURNE 

THE  MACMILLAN  CO.  OF  CANADA,  Ltd. 

TORONTO 


NAVIGATION 


BY    . 
HAROLD   JACOBY 

RUTHERFUKD    PROFESSOR    OF    ASTRONOMY 
IN    COLUMBIA    UNIVERSITY 


BOSTOK  COLLEGE  LIBRARY 
CHESTNUT  HILL,  MASS, 


SECOXD   EDITION 


ft. 


With  a  Chapter  on  Compass  Adjusting  and  a 
Collection  of  Miscellaneous  Examples 


Neto  gork 

THE   MACMILLAN   COMPANY 

1918 

All  rights  reserved 


Copyright,  1917  and  1918, 
By  the   MACMILLAN  COMPANY. 


Set  up  and  electrotyped.     Published  October,  1917. 


Second  edition,  with  new  matter,  February,  1918. 


KortDoot)  Press 

J.  S.  Gushing  Co.  —  Berwick  &  Smith  Co. 

Norwood,  Mass.,  U.S.A. 


MACLEAR   JACOBY 

QUARTERMASTER,*   THIRD   CLASS,    U.   S.   N. 

ENLISTED     FOR     THE     PERIOD     OF     THE     WAR 

THIS    VOLUME    IS    OFFERED    AS 

A     MARK     OF     RESPECT 

BY    HIS    FATHER 


*  Commissioned  Ensign,  U.  S.  N.  R.  F.,  Sbptembeb,  1917 


PREFACE 

The  present  volume  was  undertaken  with  certain  very 
definite  aims.  In  the  first  place,  it  is  intended  to  be  com- 
plete in  itself,  so  that  it  should  be  possible  to  navigate  a  ship 
in  any  ocean  not  very  near  the  north  or  south  pole  without 
other  books  or  tabular  works,  excepting  only  the  nautical 
almanac  for  the  year  in  which  the  voyage  is  made.  To  attain 
this  end  without  unduly  extending  the  size  of  the  volume, 
certain  essential  nautical  tables  have  been  abridged ;  but 
all  are  given  in  sufficiently  extended  form  to  permit  of  actual 
navigation  with  their  aid ;  and  they  are  especially  suitable 
for  beginners,  who  can  here  attain  the  necessary  knowledge 
with  less  effort  than  would  be  necessary  with  more  bulky 
volumes.  In  cases  where  very  extended  tables  are  conven- 
ient, they  are  mentioned  in  the  text. 

In  the  second  place,  the  author  has  not  assumed  that  the 
reader  possesses  formal  mathematical  and  astronomical 
knowledge,  or  desires  to  possess  such  knowledge.  When- 
ever methods  of  navigation  require  for  their  demonstration 
an  understanding  of  spherical  trigonometry,  or  some  other 
branch  of  formal  mathematical  science,  such  demonstrations 
have  been  replaced  with  incomplete  or  ''outline  "  demonstra- 
tions designed  for  the  non-mathematical  reader.  Practical 
methods  are  fully  explained ;  and  an  attempt  has  always 
been  made  so  to  word  the  explanations  that  the  reader, 
even  the  beginner,  will  understand  his  problem,  and  will 
know  what  he  is  doing,  and  why  he  does  it. 

The  requirements  of  those  who  may  study  without  a 
teacher  have  received  constant  and  special  attention.  To 
meet  these  requirements  the  whole  subject  is  presented  in 

vii 


Vlll  PREFACE 

a  somewhat  informal  manner;  such  topics  as  the  use  of 
logarithms,  or  the  principles  on  which  all  mathematical 
tables  are  constructed  —  these  less  attractive  parts  of  the 
subject  are  not  presented  in  a  special  chapter,  but  are  de- 
scribed in  a  sort  of  digression,  when  needed  in  the  discussion 
of  an  actual  navigational  problem. 

Finally,  to  further  simplify  and  condense  his  material, 
the  author  has  made  no  attempt  to  include  every  method 
that  can  possibly  be  used  to  navigate  a  ship,  or  that  ever  has 
been  used  to  navigate  a  ship ;  his  purpose  has  been  rather 
to  limit  the  volume  to  the  methods  at  present  thought  best 
by  the  most  reliable  modern  authorities. 

Other  books  on  navigation  have  been  used  freely,  espe- 
cially in  the  preparation  of  the  tables.  Among  these,  that 
admirable  encyclopedia  of  navigation,  known  as  "Bowditch," 
published  by  the  Hydrographic  Office,  United  States  Navy, 
and  Kelvin's  ''Tables  for  Sumner's  Method  at  Sea"  have 
been  found  of  the  greatest  help. 

Miss  Dorothy  W.  Block,  Instructor  of  Astronomy  in 
Hunter  College,  New  York,  has  helped  with  great  energy 
in  the  preparation  of  the  tables  and  the  correction  of  the 
text.  It  is  hoped  that  such  errors  as  may  now  remain  in 
the  book  are  few  in  number. 

H.  J. 

Columbia  University, 
August,  1917. 


PREFATORY   NOTE  TO  THE  SECOND   EDITION 

To  meet  the  wishes  of  certain  young  navigators,  this  edition 
has  an  added  chapter  on  the  adjustment  of  correctors  in  a 
compensated  compass  binnacle,  and  also  a  collection  of  new 
problems  and  examples. 

H.J. 

February,  1918. 


TABLE   OF   CONTENTS 

CHAPTER  PAGH 

I.    The  Fundamental  Problem  of  Navigation    .         .         1 

The  problem  stated.  Reasons  for  the  existence  of  the 
problem.  Definition  of  "ship's  position."  Longitude 
meridians  and  latitude  parallels.  Greenwich  the  initial 
meridian.  Position  determined  by  observation ;  on  the 
coast  and  at  sea.  Dead  reckoning.  Sextant  observa- 
tions.    Chronometer. 

II.    Dead  Reckoning  without  Logarithms     ...         7 

The  two  problems.  Designation  of  .ship's  course. 
Latitude  difference  and  departure.  The  traverse  table. 
Use  and  construction  of  tables  in  general.  Arguments 
and  tabular  numbers.  Relation  between  departure  and 
longitude  difference.     Middle  latitude. 

III.  Dead  Reckoning  with  Logarithms  ...       23 

Explanation  of  number  logarithms  and  their  use. 
Multiplication  and  division.  Trigonometric  logarithms. 
Solution  of  the  two  problems.  Middle  latitude  sailing. 
Mercator  sailing.  Meridional  parts.  Great  circle  sail- 
ing. The  rhumb  line.  Composite  sailing.  Parallel 
sailing.     Traverse  sailing. 

IV.  The  Compass 40 

The  card,  how  divided.     Degrees  and  points.     Boxing 

the  compass.  Lubber  line.  True  course  and  compass 
course.  Error,  variation  and  deviation.  Swinging  ship. 
Azimuth  circle  and  pelorus.  The  compass  formulas. 
The  two  deviation  tables.  Comparative  table  of  points 
and  degrees. 

V.    Coastwise  Navigation       ......       53 

The  "fix."  Bow  bearings.  Doubling  the  bearing  on 
the  bow\  Bow  and  beam  bearings.  Distance  a-beam. 
Cross  bearings.  The  danger  angle.  Danger  bearing. 
Soundings. 

ix 


X  TABLE   OF   CONTENTS 

CHAPTER  PAGE 

VI.    The  Sextant       .  61 

Description  of  the  instrument  and  its  use.  The  vernier. 
Index  error.  Three  adjustments.  The  artificial  horizon. 
Correcting  the  altitude.     Dip.     Refraction.     Parallax. 

VII.   The  Nautical  Almanac     ......       75 

Specimen  pages  of  it.  Greenwich  mean  time.  Decli- 
nation. Equation  of  time.  Astronomic  and  civil  day. 
Apparent  solar  time.  Chronometers  and  the  rate  card. 
Right  ascension.     Solar  and  sidereal  time. 

VIII.    Older  Navigation  Methods     .....       86 

The  noon-sight  for  latitude.  Tropic  observations  and 
the  midnight  sun  in  high  latitudes.  Preparing  for  the 
observation.  Setting  the  cabin  clock.  Star  observa- 
tion. Ex-meridian  observation.  The  time-sight  for 
longitude.  Set  of  current.  Star  time-sight.  Condensed 
forms  of  calculation. 

IX.   Newer  Navigation  Methods    .....     108 

Errors  produced  by  dead  reckoning.  Captain  Sumner, 
and  the  Sumner  line.  Bearing  of  the  line.  The  Sumner 
point.  Azimuth  tables.  Condensed  form  of  calcula- 
tion. Star  observations.  Comparison  of  Sumner  navi- 
gation with  time-sight  navigation.  The  Kelvin  table. 
Condensed  forms  of  sun  and  star  observations.  Inter- 
section of  two  Sumner  lines  obtained  with  a  special  table. 
Motion  of  ship  between  observations. 

X.  A  Navigator's  Day  at  Sea  .  .  .  .  .  141 
Voyage  planned  from  New  York  to  Colon.  Departure 
at  Sandy  Hook  lightship.  The  course  to  Watlings  Island. 
The  variation  and  deviation  applied.  Azimuth  of  the  sun 
observed  at  sunrise.  Bow  and  beam  bearings  of  Barnegat 
Light.  The  patent  log  and  the  log  book.  New  course 
from  Barnegat.  Morning  sight  worked  as  a  Sumner  line. 
Another  Azimuth  observation.  Weather  thickens  at 
11 :  30.  Ex-meridian  sight  at  11 :  42,  worked  as  a  Sumner 
line.  Afternoon  sight  worked  as  a  Sumner  line.  Posi- 
tion of  ship  fixed  from  intersection  of  the  two  lines.  East- 
erly current  estimated.  Compass  error  again  tested. 
The  course  set  for  the  night. 

Tables  ............     153 

Appendix  1.     Compass  Adjusting  .....     323 

Appendix  2.    Miscellaneous  Examples  .....     335 


LIST   OF  ABBREVIATIONS 

Used  in  the  Present  Volume 

Alt.  for  altitude ; 

App.  for  apparent ; 

Arg.  diff.  for  argument  difference ; 

Cf .  for  compare  ; 

Chron.  for  chronometer ; 

Comp'd  for  computed; 

Cos  for  cosine ; 

Cot  for  cotangent ; 

Csc  for  cosecant ; 

C.  —  W.  for  chronometer  minus  watch ; 
Dec.  for  declination ; 

Dep.  for  departure ; 

Dist.  for  distance ; 

D.  R.  for  dead  reckoning; 
Eq.  for  equation  of  time  ; 

G.  A.  T.  for  Greenwich  apparent  time; 

G.  M.  T.  for  Greenwich  mean  time ; 

Hav.  for  haver  sine ; 

H.  D.  for  hourly  difference  ; 

Int.  diff.  for  interpolation  difference ; 

Lat,  for  latitude ; 

Lat.  diff.  for  latitude  difference ; 

Log  for  logarithm ; 

Long.  for  longitude ; 

Long.  diff.        for  longitude  difference ; 

Mer.  lat.  diff.  for  meridional  latitude  difference ; 

Obs'd  for  observed ; 

p  for  polar  distance ; 

R.  A.  for  right  ascension ; 

s  for  half  sum ; 

Sec  for  secant ; 

Sin  for  sine ; 

T  for  ship's  apparent  solar  time  (or  star's  hour-angle) ; 

Tab.  diff.  for  tabular  difference ; 

Tan  for  tangent. 

xi 


NAVIGATION 

CHAPTER   I 
THE  FUNDAMENTAL  PROBLEM  OF  NAVIGATION 

To  find  one's  way  in  a  ship  across  the  trackless  ocean  is 
our  problem.  Most  people  would  like  to  know  how  it  is 
solved ;  nor  is  the  solution  very  difficult  to  understand  when 
set  forth  in  simple  language  and  without  too  great  wealth  of 
technical  detail.  We  hope  the  reader  will  find  this  to  be 
the  case  after  a  study  of  the  following  pages. 

Our  fundamental  problem  can  be  more  fully  stated  quite 
easily.  It  consists  in  the  determination  of  a  ship's  location 
on  the  earth's  surface  at  any  given  moment.  If  this  loca- 
tion can  be  determined,  it  becomes  a  comparatively  easy 
matter  to  ascertain  the  direction  (north,  south,  northeast, 
southeast,  etc.)  in  which  the  ship  must  be  steered  in  order 
to  reach  her  port  of  destination.  For  the  location  of  the 
port  of  destination  on  the  earth's  surface  is  of  course  also 
known  :  and  if  we  know  where  the  ship  and  her  destined  port 
both  are,  we  can  easily  find  the  right  course  for  the  helmsman. 

With  the  fundamental  problem  stated  in  this  way,  it 
would  almost  seem  as  if  there  were  really  no  such  problem 
in  existence.  For  when  the  ship  begins  her  voyage,  she  is 
necessarily  in  a  known  port.  Knowing  also  the  port  to 
which  she  is  to  go,  we  should  be  able  to  determine  her  proper 
course  from  the  one  known  port  to  the  other.  This  course 
being  then  steered,  no  further  navigational  proceedings  would 
be  required.     But  this  reasoning  is  incorrect,  because  a  ship 

B  1 


2  NAVIGATION 

does  not  actually  advance  across  the  ocean  in  exactly  the 
direction  in  which  she  is  steered.  Ocean  currents  deflect 
her ;  and  the  action  of  a  strong  wind  blowing  against  one  of 
her  sides  will  have  a  similar  effect.  Currents  and  winds 
cannot  be  predicted  with  accuracy :  and  so  it  becomes 
necessary  to  re-determine  the  ship's  position  frequently  at 
sea.  This  should  be  done  at  least  once  daily  if  possible; 
and  when  it  has  been  done,  the  mariner  can  take  a  new 
*' departure/'  as  he  calls  it,  and  lay  a  new  course  for  his 
intended  port.  Thus  the  effect  of  ocean  currents,  etc.,  can 
be  eliminated,  and  the  voyage  made  as  safely  as  if  they  did 
not  exist. 

Now  this  determination  of  the  ship's  position  at  sea, 
and  when  out  of  sight  of  land,  is  strictly  an  astronomical 
problem.  It  can  be  solved  by  means  of  astronomical  ob- 
servations, and  in  no  other  way.  But  before  giving  an  out- 
line of  how  this  is  done,  let  us  first  see  what  is  meant  by 
the  words  ''ship's  position  at  sea."  How  can  we  describe 
a  ship's  position  so  that  one  mariner  could  tell  another 
where  she  is  located,  and  thus  enable  the  second  mariner  to 
find  her? 

To  thus  indicate  the  point  on  the  earth's  surface  occupied 
by  the  ship  has  a  certain  similarity  with  giving  the  address  of 
a  house  in  a  city.  Such  a  city  address  always  consists  of 
two  separate  statements;  as,  for  instance,  the  name  of  a 
street  and  the  number  of  the  house.  An  address  cannot 
be  given  completely  unless  two  different  facts  are  stated. 
They  need  not  necessarily  be  a  street  name  and  a  street 
number :  we  can  equally  well  designate  such  an  address  by 
stating  that  the  house  is  at  the  corner  of  a  certain  street  and 
a  Certain  avenue.  But  here  also  the  address  is  made  up  of 
two  separate  facts. 

This  form  of  stating  an  address  as  the  intersection  of  a 

certain  street  and  avenue  is  the  form  having  the  closest 

^resemblance  to  the  method  of  the  navigator.     If  the  city 

avenues  are  supposed  to  run  north  and  south,  and  the  streets 


THE   FUNDAMENTAL  PROBLEM  OF  NAVIGATION     3 

east  and  west,  as  they  do  in  New  York  (approximately),  the 
analogy  with  navigation  will  be  almost  perfect. 

For  the  navigator  imagines  the  earth  covered  with  a  net- 
work consisting  of  ''avenues, "  running  north  and  south,  and 
''streets,"  running  east  and  west.  He  calls  the  "avenues" 
meridians  of  longitude,  and  the  "streets"  parallels  of  latitude. 
Then  he  designates  the  position  of  a  ship  on  the  ocean  by 
stating  that  it  is  at  the  intersection  of  a  certain  meridian 
of  longitude  and  parallel  of  latitude.  There  are  360  such 
meridians  of  longitude  :  each  begins  at  the  terrestrial  equator, 
and  runs  north  and  south  from  there  to  the  north  and  south 
poles  of  the  earth.  Of  the  latitude  parallels  there  are  180.^ 
They  all  run  east  and  west,  parallel  to  the  terrestrial  equator ; 
90  are  between  the  equator  and  the  north  pole,  and  the  other 
90  between  the  equator  and  the  south  pole. 

One  of  the  longitude  meridians  (that  passing  through 
Greenwich,  England)  is  chosen  arbitrarily  as  the  starting 
point  for  counting  longitude  meridians.  To  this  initial 
meridian  is  assigned  the  number  0,  and  the  other  meridians 
are  numbered  successively  1,  2,  3,  etc.  So  numbered, 
the  meridians  are  called  "degrees"  of  longitude;  the  third 
one,  for  instance,  being  written  3°.  The  meridians  may  be 
counted  either  eastward  or  westward  from  Greenwich,  a 
ship  on  the  20th  meridian  west  of  Greenwich,  for  instance, 
being  in  longitude  20°  west. 

The  latitude  parallels  are  similarly  counted  north  and 
south  from  the  equator ;  and  if  the  above  ship  were  on  the 
40th  latitude  parallel  north  of  the  equator,  her  complete 
"address,"  or  position  at  sea,  would  be  long.  20°  W. ;  lat. 
40°  N. 

Of  course  a  ship  would  only  rarely  be  located  exactly  at 
the  intersection  of  a  meridian  and  parallel.  Therefore,  the 
space  between  any  two  successive  meridians  and  between 
any  two  successive  parallels  is  subdivided  into  60  parts, 
called  minutes  of  arc.      Thus  the  above  ship,  if  halfway 

*  Including  the  equator  twice,  but  excluding  the  two  poles. 


4  NAVIGATION 

between  a  pair  of  meridians  and  also  halfway  between  a 
pair  of  parallels,  might  be  in  longitude  20°  30'  west,  and 
in  latitude  40°  30'  north.  This  would  be  written  long. 
20°30'W.;  lat.  40°30'N. 

Each  minute  of  longitude  and  latitude  is  further  sub- 
divided, when  extreme  accuracy  is  required,  into  60  seconds ; 
so  that  if  the  ship  were  a  little  to  the  north  and  a  little  to 
the  west  of  the  above  position,  she  might,  for  instance,  be 
in  long.  20°  30'  26"  W. ;  lat.  40°  30'  10"  N. 

These  meridians  and  parallels,  or  longitude  and  latitude 
lines,  appear  on  many  maps  and  charts  as  straight  lines, 
or  at  least  as  lines  only  slightly  curved.  But  being  all  lines 
imagined  drawn  on  the  earth,  which  is  almost  an  exact 
sphere  or  round  ball,  they  must  really  all  be  circles.  Thus, 
the  terrestrial  equator  is  really  a  big  circle,  girdling  the 
earth,  and  divided  into  360  equal  parts,  or  degrees.  At 
each  of  the  division  points  a  meridian  starts  northward 
toward  the  pole.  This  meridian  is  also  a  big  circle 
perpendicular  to  the  equator.  The  distance  along  the 
meridian  from  the  equator  to  the  pole  is  divided  into  90 
equal  parts  or  degrees,  and  the  whole  distance  from  equator 
to  pole  is  one  quarter  of  a  complete  circumference  of  the 
earth.  The  90  degrees,  from  equator  to  pole,  thus  repre- 
senting one  quarter  of  a  circumference  of  the  earth,  a  com- 
plete circumference  contains  4  X  90,  or  360  degrees,  the 
same  as  the  equator.  So  the  degrees  measured  along  the 
meridians  are  equal  to  the  degrees  measured  along  the 
equator.  The  former  are  degrees  of  latitude,  the  latter 
degrees  of  longitude ;  and  degrees  of  latitude  are  equal  to 
degrees  of  longitude,  when  the  latter  are  measured  along 
the  equator.  The  length  of  each  degree  is  then  60  nautical 
miles. 

Having  thus  indicated  what  is  meant  by  a  ship's  position 
in  latitude  and  longitude,  we  shall  next  describe  in  outline 
how  such  a  position  may  be  determined  by  observation. 
If  the  ship  is  within  sight  of  a  coast-line,  there  will  probably 


THE  FUNDAMENTAL  PROBLEM  OF  NAVIGATION     5 

be  some  lighthouse,  or  other  ''aid  to  navigation,"  in  view, 
from  which  the  navigator  can  ascertain  where  he  is.  Methods 
for  doing  this  are  described  later  (p.  53).  But  when  the 
ship  is  really  at  sea,  with  no  land  in  sight,  real  deep-sea 
methods  must  be  employed. 

These  methods,  when  the  weather  is  clear,  always  include 
an  observation  of  the  sun  or  some  other  heavenly  body. 
When  the  weather  does  not  permit  such  observations,  the 
mariner  can  still  find  his  position  approximately  by  means 
of  ''dead  reckoning"  (abbreviated,  D.  R.).  This  process 
will  be  described  in  detail  in  the  next  chapter ;  but  we  can 
already  state  that  it  consists  in  a  calculation  based  on  his 
astronomic  observation  of  latest  date.  Knowing  where  the 
ship  was  the  last  time  he  observed  the  sun,  and  also  know- 
ing both  the  direction  in  which  he  has  steered  and  the 
(approximate)  speed  of  the  ship,  the  navigator  can  calculate 
(also  approximately)  the  location  of  the  point  he  has  reached. 

Even  when  astronomical  observations  are  made,  the 
D.  R.  calculation  is  always  carried  out,  because  the  navi- 
gator is  always  anxious  to  know  how  nearly  correct  his 
D.  R.  result  would  have  been,  if  the  day  had  been  cloudy. 
Furthermore,  this  result  also  acts  as  a  check  on  the  astronomi- 
cal work,  and  tejids  to  increase  the  navigator's  confidence 
in  the  correctness  of  his  final  result  as  to  the  ship's  location. 

The  manner  in  which  the  ship's  position  is  found  from 
astronomic  observations  will  of  course  be  explained  in  detail 
later.  It  is  all  done  with  an  instrument  called  a  sextant. 
This  is  merely  a  contrivance  with  which  the  navigator  can 
measure  how  high  the  sun  (or  other  heavenly  body)  is  in  the 
sky  at  any  moment.  The  sun  is  highest  in  the  sky  daily 
at  noon,  but  it  is  not  equally  high  on  different  days  in  the 
year.  Nor  is  it  equally  high  on  the  same  date  in  different 
latitudes.  Thus,  by  measuring  with  the  sextant  how  high 
it  is  on  any  particular  date  at  noon,  as  seen  from  the  ship, 
the  navigator  learns  the  terrestrial  latitude  in  which  the 
ship  is  located. 


6  NAVIGATION 

Similar  sextant  observations  made  at  other  suitable  times 
during  the  day,  when  combined  with  exact  readings  taken 
from  an  accurate  chronometer  such  as  every  ocean-going 
ship  carries,  will  similarly  make  the  ship's  longitude  known. 
All  this  will  of  course  be  explained  in  full  detail  in  later 
chapters. 


CHAPTER  II 


DEAD   RECKONING   WITHOUT   LOGARITHMS 

As  we   have   seen  (p.  5),  this  is  a  process   by  means  of 
which  the  mariner  can  calculate  a  ship's  position  in  latitude 


a* 


60° 


West  Longitude 
59'  58°  57° 


56* 


55' 


46 


45° 


44 


43 


42 


41 


40 


Fig.  1.  —  Dead  Reckoning.     (Diagram  not  drawn  to  scale.) 

7 


8  NAVIGATION 

and  longitude,  without  special  astronomic  observations  of 
any  kind.  In  the  accompanying  Fig.  1,  which  represents  a 
portion  of  a  chart  of  the  North  Atlantic,  a  ship's  position 
at  noon  is  shown  at  the  point  Y.  This  point  we  will  call 
the  ship's  'initial  position,"  in  discussing  our  present  prob- 
lem. We  will  suppose  that  it  was  correctly  obtained  by  as- 
tronomic observations,  and  that  these  showed  the  ship  at  Y 
to  be  in  lat.  42°  11'  N.  and  long.  59°  28'  W.  from  Green- 
wich. Sometime  in  the  afternoon,  having  traveled  a  dis- 
tance estimated  from  the  known  speed  of  the  ship  as  63  miles, 
and  having  ''made  good"  this  distance  in  the  direction  YP, 
the  ship  arrives  at  P.  This  point  P  we  will  call  the  ship's 
''final  position"  ;  and  our  problem  now  is  to  find  its  latitude 
and  longitude. 

This  problem  may  be  called  the  first  fundamental  dead- 
reckoning  problem.  The  second  and  remaining  fundamental 
problem  is  the  converse  of  the  first,  and  may  be  stated  as 
follows  :  having  given  the  latitude  and  longitude  of  the  initial 
point  F,  as  occupied  by  the  ship,  and  also  the  latitude  and 
longitude  of  the  final  point  P,  it  is  required  to  find  the  dis- 
tance from  F  to  P  in  miles,  and  also  the  direction  of  the  line 
YP.^ 

To  understand  these  two  problems  properly  it  is  next 
necessary  to  explain  how  we  may  define  the  words  "direc- 
tion FP."  This  is  done  by  referring  the  line  YP  to  the 
direction  of  the  arrow  shown  in  the  figure.  This  arrow 
is  parallel  to  the  longitude  meridians  on  the  chart,  and 
therefore  points  due  north.  The  angle  between  the  arrow 
YN  and  the  line  YP  is  marked  in  the  figure,  and  is  called 
the  "ship's  course."  This  angle  is  really  the  difference  in 
direction  of  the  two  lines  FA^"  and  YP.  The  point  F  is  called 
the  "vertex"  of  the  angle,  and  all  angles  are  designated 

*  We  think  it  advisable  to  place  these  two  important  converse 
problems  together,  and  to  call  them  both  problems  of  dead  reckon- 
ing, though  many  writers  on  navigation  confine  the  phrase  *'  dead 
reckoning"  to  the  first  fundamental  problem  alone. 


DEAD   RECKONING  WITHOUT   LOGARITHMS 


9 


Y 

Fig.  2.  — Dead 
Reckoning. 


by  three  letters,  the  letter  belonging  to  the  vertex  being 
placed  between  the  other  two ;  in  this  case  the  angle  is 
called  either  NYP  or  PYN. 

Now  let  us  draw  a  line  PQ  (fig.  2),  from  P  to  NY,  and 
perpendicular  to  NY.  Then  the  motion  of  the  ship  from 
F  to  P  will  have  carried  her  north  of  the  i^j^ 
point  F  by  a  distance  equal  to  YQ,  and  east 
of  the  point  F  by  a  distance  equal  to  QP. 
This  is  not  strictly  true,  unless  the  earth's 
surface,  throughout  the  small  area  involved 
in  the  present  problem,  can  be  regarded 
as  a  fiat  surface.  Such  a  flat  surface  is 
called  in  geometry  a  ''plane"  surface;  and 
these  calculations  therefore  belong  to  that 
part  of  navigation  which  is  called  ''plane  sailing."  Plane- 
sailing  calculations  are  easy  calculations,  and  they  are 
generally  sufficiently  accurate  for  the  purposes  of  the 
navigator. 

The  ship's  course,  being  thus  an  angle,  must  be  designated 

by  means  of  a  unit  of  measure 
suitable  for  measuring  angles. 
For  this  purpose  the  degrees  and 
minutes  already  used  for  longi- 
tude   and    latitude   (p.  3)    are 
usually  employed.    Fig.  3  shows 
that  a  latitude,  for  instance,  is 
really  an  angle,  and  must  there- 
fore also   be   measured  in   de- 
grees.   P  is  the  earth's  pole,  PQ 
a  meridian,  and  the  latitude  of 
the  observer  at  0  is  the  angle 
OCQ,  here  about  40°. 
So  it  is  clear  that  the  ship's  course  NYP  (figs.  1  and  2) 
will  be  measured  in  degrees.     Minutes  are  not  really  needed 
in  measuring  courses,  as  they  are  in  measuring  latitudes; 
the  nearest  whole  degree  is  always  accurate  enough,  because 


Fig.  3.  —  Latitude  Angle. 


10  NAVIGATION 

it  is  never  possible  to  steer  a  ship  on  her  proper  course  with 
absolute  exactness.  In  fact,  many  mariners  use  a  still  less 
precise  method  of  measuring  courses  by  means  of  'Hhe  points 
of  the  compass."     (See  p.  40.) 

Resuming  our  two  fundamental  problems  (p.  8),  let  us 
now  begin  with  the  first  one,  and  proceed  to  find  the  lati- 
tude and  longitude  of  the  point  P  (figs.  1  and  2).  To  solve 
this  problem,  we  must  not  only  know  the  distance  YP 
(63  miles),  as  traveled  by  the  ship,  but  also  the  number  of 
degrees  in  the  course  angle  NY  P.  Let  us  suppose  this  course 
angle  happens  also  to  be  40°.  The  problem 
then  appears  as  shown  in  Fig.  4.  We  now 
know  the  distance  YP  and  the  angle  QYP. 
Evidently  the  next  step  is  to  find  the  distances 
QFand  QP.  QY,  in  our  present  problem,  is 
called  a  'latitude  difference"  and  QP  is  called 
Y^y  a  ''departure." 

Fig.  4.  — Dead  To  find  the  ''latitude  difference"  and 
ning.  '^ departure"  from  the  course  angle  and  dis- 
tance we  may  either  use  that  branch  of  mathematics  called 
plane  trigonometry,  or  we  may  find  them  from  a  special 
navigation  table,  called  a  "traverse  table."  Our  Table  1 
(beginning  p.  154)  is  such  a  table. 

Before  ^  beginning  its  use  it  will  be  well  for  the  reader  to 
note  in  general  that  all  mathematical  tables  consist  of  two 
sets  of  numbers.  The  first  set  of  numbers  are  called  "argu- 
ments" of  the  table,  and  the  second  set  are  called  "tabular 
numbers."  The  main  object  of  the  table  is  to  furnish  us 
w^ith  the  proper  tabular  number  when  we  know  the  proper 
argument. 

The  ordinary  multiplication  table  is  a  good  example  of  a 
mathematical  table.     It  is  usually  written  as  follows  and 

1  The  beginner  may  find  it  advisable,  on  a  first  reading  of  the 
book,  to  omit  this  explanation  of  mathematical  tables,  returning 
later  when  he  finds  a  reference  to  it  in  the  text.  The  dead  reckoning 
problem  under  discussion  is  resumed  on  p.  13. 


DEAD  RECKONING  WITHOUT  LOGARITHMS       11 

it  affords  a  good  opportunity  of  studying  the  principles 
underlying  all  mathematical  tables  in  a  case  so  simple  as 
to  offer  no  difficulty. 

MULTIPLICATION    TABLE 
(to  illustrate  "argument"  and  "  tabular  number") 


2 

2 

3 

3 

4 
4 

6 

5 

6 

7 

8 

9 

10 

11 

12 

1 

6 

7 

8 

9 

10 

11 

12 

2 

4 

6 

8 

10 

12 

14 

16 

18 

20 

22 

24 

3 

6 

9 

12 

15 

18 

21 

24 

27 

30 

33 

36 

4 

8 

12 

16 

20 

24 

28 

32 

36 

40 

44 

48 

6 

10 

15 

20 

25 

30 

35 

40 

45 

50 

55 

60 

6 

12 

18 

24 

30 

36 

42 

48* 

54 

60 

66 

72 

7 

14 

21 

28 

35 

42 

49 

56 

63 

70 

77 

84 

8 

16 

24 

32 

40 

48 

56 

64 

72 

80 

88 

96 

9 

18 

27 

36 

45 

54 

63 

72 

81 

90 

99 

108 

10 

20 

30 

40 

50 

60 

70 

80 

90 

100 

110 

120 

11 

22 

33 

44 

55 

66 

77 

88 

99 

110 

121 

132 

12 

24 

36 

48 

60 

72 

84 

96 

108 

120 

132 

144 

In  this  table  the  arguments  are  printed  in  heavy  type  and 
are  contained  in  the  left-hand  column  and  the  topmost 
horizontal  line.  In  using  the  table,  these  arguments  are 
given  in  pairs,  being  always  the  pair  of  numbers  to  be  mul- 
tiplied. In  fact,  in  the  case  of  most  tables,  the  arguments 
are  thus  given  in  pairs,  though  there  are  some  tables  with 
but  a  single  argument.  In  the  present  case  one  number 
from  the  pair  of  arguments  will  be  found  in  the  left-hand 
column,  the  other  in  the  top  horizontal  line.  Thus,  if  we  wish 
to  multiply  6  and  8,  these  two  numbers  constitute  the  pair 
of  arguments.  We  find  the  right  line  (belonging  to  6)  and 
column  (belonging  to  8) ,  and  the  tabular  number  48  (marked 
with  a  *)  occurs  at  the  intersection  of  the  6-line  and  the  8- 
column.  If  the  pair  of  arguments  are  taken  in  the  order 
8x6  instead  of  6  X  8,  we  should  use  the  8-line  and  the 
6-column,  again  finding  the  required  product  (48)  as  the 
tabular  number  at  the  intersection. 


12  NAVIGATION 

Sometimes  the  given  arguments  cannot  be  found  di- 
rectly in  the  table.  Thus  we  might  wish  to  multiply 
6§  (written  6.5)  by  8.  Evidently  the  proper  tabular 
number  would  be  halfway  between  the  6x8  tabular 
number  (48)  and  the  7x8  tabular  number  (56).  The 
correct  answer  would  therefore  be  52.  This  process,  by 
which  the  tabular  number  52  is  obtained,  is  called  ''in- 
terpolation." The  example  6|  X  8  is  an  extremely  simple 
one.  When  less  easy  ones  occur,  the  interpolation  is  best 
made  as  follows :  we  ascertain  by  subtraction  how  much 
the  tabular  number  increases  while  the  argument  changes 
from  6  to  7.  This  increase  is  here  8,  because  the  tabular 
number  changes  from  48  to  56  in  the  8-column,  while  the 
argument  in  the  left-hand  column  changes  from  6  to  7. 
This  increase  of  8  in  the  tabular  number  is  called  a  ''tabular 
difference."  We  now  compare  the  given  argument  (6.5) 
with  the  nearest  argument  (6)  occurring  in  the  left-hand 
column  of  arguments,  and  find  an  "argument  difference" 
of  0.5  (being  6.5  minus  6).  Since  this  "argument  dif- 
ference" is  0.5,  we  must  evidently  take  0.5  X  8  (8  being  the 
tabular  difference),  and  increase  the  tabular  number  48  by 
0.5  X  8,  or  4.  This  again  brings  us  to  52.  Similar  exam- 
ples are : 

(1)  5.3  X  4  =  21.2;    (2)  7.7  X  8  =  61.6. 

In  example  (1)  the  tabular  numbers  are  20  and  24 ;  the 
tabular  difference  is  4.  0.3  X  4  =  1.2;  20  +  1.2  =  21.2,  the 
answer.  Both  examples  may  be  verified,  of  course,  by  ordi- 
nary multiplication. 

When  both  given  arguments  contain  fractions,  as,  for 
instance,  5.3  X  8.4,  the  resulting  "double  interpolation" 
is  so  complicated  as  to  be  of  little  practical  use  to  the  navi- 
gator. 

To  make  this  general  explanation  of  mathematical  tables 
complete,  it  remains  to  show  how  they  can  be  used  in  an 
inverse  manner ;  i.e.  to  find  the  argument  from  the  tabular 


DEAD  RECKONING  WITHOUT  LOGARITHMS       13 

number.  Thus,  if  we  were  told  that  the  tabular  number  is 
48,  and  one  argument  8,  an  inspection  of  the  table  would 
at  once  show  that  the  other  argument  must  be  6.  In  this 
way  the  table  might  be  used  for  division  as  well  as  multi- 
plication ;  and  interpolation  would  evidently  also  be  possible. 
Many  mathematical  tables  must  frequently  be  thus  used 
in  an  inverse  manner. 

Having  thus  explained  the  peculiarities  of  mathematical 
tables,  we  return  to  our  dead-reckoning  problem  and  its 
solution  by  means  of  the  traverse  table  (p.  154). 

Referring  to  that  table  we  find  a  column  (p.  167), 
headed  40°,  the  course  angle  of  our  present  problem.  On 
the  left-hand  side  of  the  page  we  find  the  given  distance,  63. 
Then,  opposite  the  distance  63,  and  under  40°,  we  find  the 
latitude  difference  (abbreviated,  ''Lat. ")  and  the  departure 
(abbreviated,  '^Dep.")  to  be: 

lat.  =  48.3,  dep.  =  40.5. 

The  following  are  additional  examples  for  practice : 

Given :  dist.,  84,  course  26° ;  Ans.y  lat.  =  75.5,  dep.  =  36.8. 
Given :  dist.,  28,  course  11° ;  Ans.,  lat.  =  27.5,  dep.  =   5.3. 

When  the  course  is  between  1°  and  45°  the  course  angle 
will  be  found  in  Table  1  at  the  head  of  the  column  :  but  when 
the  course  is  between  45°  and  90°,  it  appears  at  the  foot  of 
the  column.  In  the  latter  case,  the  tabular  lat.  and  dep. 
are  to  be  taken  from  the  columns  having  " Lat. "  and  ''Dep." 
at  the  foot  instead  of  the  top  of  the  column.  Examples 
follow : 

Given :  dist.,  63,  course  50° ;  Ans.,  lat.  =  40.5,  dep.  =  48.3. 
Given :  dist.,  84,  course  64°  ;  Ans.,  lat.  =  36.8,  dep.  =  75.5. 
Given :  dist.,  28,  course  52° ;  Ans.,  lat.  =  17.2,  dep.  =  22.1, 

In  addition  to  the  course  angles  from  1°  to  90°,  three  ad- 
ditional angles  are  given  in  parentheses  at  the  top  and  foot 
of  each  column.  Thus,  with  the  course  angle  30°  appear 
also  150°,  210°,  330°.     This  simply  means  that  the  latitudes 


14 


NAVIGATION 


Fig.  5.  — De- 
partures for 
30°  and  150°. 


and  departures  are  the  same  for  these  four 
course  angles.  The  accompanying  Fig.  5  shows, 
for  instance,  that  the  departures  QP  and  Q'P' 
are  equal  for  30°  and;  150°  courses  if  the  two 
distances  YP  and  YP'  are  alike. 

It  will  be  noticed  also  that  our  traverse  table 
always  gives  distances  from  1  to  50  on  a  left- 
hand  page,  and  from  50  to  100  on  a  right-hand 
page.  When  distances  larger  than  100  occur, 
it  is  necessary  to  use  the  100,  200,  etc.,  given  on 
the  lower  part  of  each  page.  If,  for  instance, 
we  require  the  latitude  and  departure  for  a 
distance  363  miles,  course  40°,  we  turn  again  to 
the  40°  column,  and  find  (near  the  bottom  of 
the  page) : 

For  300  miles,  lat.  =  229.8,  dep.  =  192.8 
and  (in  the  usual  way)  for  63  miles,  lat.  =   48.3,  dep.  =   40.5 
Sums,  363  =278.1  233.3 

Consequently,  for  dist.  363,  course  40°,  lat. =278.1,  dep. =233.3. 

Other  examples  are : 

Course  25°,  dist.,  452 ;  lat.  =  409.6,  dep.  =  191.0. 
Course  68,  dist.,  521 ;  lat.  =  195.2,  dep.  =  483.1. 
Course  226,    dist.,  384 ;  lat.  =  266.8,  dep.  =  276.2. 

When  the  given  distances  or  course  angles,  which  are 
really  the  ''pairs  of  arguments"  (p.  11)  of  the  traverse  table, 
contain  fractions,  interpolation  can  be  used ;  but  such  close 
accuracy  is  seldom,  if  ever,  required  in  navigation. 

More  extended  traverse  tables  will  be  found  in  Bowditch's 
*' American  Practical  Navigator,"  published  by  the  Navy 
Department,  Washington.  They  are  also  printed  separately 
in  Bowditch's  ''Useful  Tables."  Both  volumes  can  be 
purchased  at  any  "navigation  shop"  where  instruments 
and  books  suitable  for  navigators  are  sold. 

To  complete  this  explanation  of  our  traverse  table,  it  is 
still  necessary  to  mention  that  it  also  provides,  with  suf- 
ficiently close  approximation,  for  the  method  of  measuring 


DEAD  RECKONING  WITHOUT  LOGARITHMS       15 

course  angles  in  *' points  of  the  compass"  (pp.  10,  41).  This 
method  is  not  now  in  use  in  the  United  States  Navy,  but  it 
is  still  largely  employed  in  merchant  vessels.  It  is  sufficient 
to  state  here  that  a  course  of  3  points,  for  instance,  is  very 
nearly  equal  to  a  course  of  34°,  and  the  traverse  table  column 
for  34°  may  properly  be  used  for  a  3-point  course.  Similarly, 
31°  may  be  used  for  2f  points,  and  the  mariner  desiring  to  use 
points  can  always  find  from  the  traverse  table  itself  just 
what  column  to  use.  A  special  traverse  table  for  points  may 
also  be  found  in  Bowditch's  Tables,  already  mentioned. 

We  have  now  shown  how  to  find  latitude  difference  and 
departure  by  means  of  the  traverse  table.  But  our  problem 
is  not  yet  completely  solved.  Our  ship  (p.  8)  started  from 
the  point  Y  in  lat.  42°  lU  N. ;  long.  59°  28'  W.  She  traveled 
63  miles  on  a  40°  course,  and  the  traverse  table  showed  that 
she  thus  made  good  a  latitude  difference  of  48.3  miles  and  a 
departure  of  40.5  miles.  It  now  remains  to  ascertain  how 
much  the  ship  changed  her  latitude  in  degrees  and  minutes 
from  42°  lU  N.  and  her  longitude  in  degrees  and  minutes 
from  59°  28'  W.  When  we  have  found  these  last  changes, 
we  can  learn  the  latitude  and  longitude  of  the  point  P, 
which  we  are  required  to  find. 

To  get  the  latitude  change  in  degrees  and  minutes  from 
the  latitude  difference  in  miles  offers  no  difficulty.  If  the 
miles  used  are  nautical  miles  (and  in  navigation  they  always 
are  nautical  miles),  each  mile  of  latitude  difference  corre- 
sponds to  1'  of  angular  measure  (p.  9),  and  60  miles  corre- 
spond to  1°.  Thus  our  ship  must  have  changed  her  latitude 
48'. 3,  corresponding  to  a  latitude  difference  of  48.3  miles. 
Her  initial  latitude  having  been  42°  lU  N.,  her  final  latitude 
at  P  will  be  42°  lU  +  48'  (if  we  omit  the  odd  .3)  or  42°  59'  N. 

The  relation  between  departure  and  difference  of  longitude 
is  not  quite  so  simple.  Our  ship's  departure  of  40.5  miles 
might  correspond  to  far  more  than  40.5  minutes  of  longitude. 
In  fact,  in  very  high  latitudes  near  the  north  pole,  the  longi- 
tude meridians  converge  so  closely  that  a  person  traveling 


16  NAVIGATION 

a  few  miles  might  change  his  longitude  very  greatly.  At  the 
pole  itself  a  man  might  change  his  longitude  180°  by  simply 
stepping  across  the  pole.  So  it  follows  that  the  longitude 
difference  in  minutes  is  greater  than  the  departure  in  miles 
(however,  cf .  p.  4) .  The  difference  between  the  two  increases 
rapidly  as  we  approach  high  latitudes  though  it  is  nil  at 
the  equator ;  in  Table  2  (beginning  p.  168)  we  give  this 
excess  of  longitude  difference  over  departure  for  all  latitudes 
under  60°,  and  for  all  longitude  differences  up  to  100.  When 
the  longitude  differences  are  greater  than  100,  it  is  necessary 
to  use  the  numbers  given  for  100,  200,  300,  etc.,  near  the 
bottom  of  each  page  in  the  table,  and  to  sum  tabular  num- 
bers, precisely  as  we  did  with  the  traverse  table. 

It  will  be  noticed  that  Table  2  gives  '^ tabular  numbers" 
for  each  degree  of  latitude  in  a  separate  column,  and  that 
these  various  latitudes  are  called  '^ middle  latitudes."  Thus 
the  middle  latitude  and  the  longitude  difference  are  the  pair 
of  arguments  (p.  11)  for  Table  2,  and,  as  we  shall  see  pres- 
ently, the  use  of  the  middle  latitude  avoids  any  uncertainty 
in  choosing  the  correct  column  for  use.  In  our  present 
problem  we  have  at  our  disposal  (p.  15)  two  different  lat- 
itudes :  the  initial  latitude  at  the  point  F,  42°  11'  N.,  and 
the  final  latitude  at  the  point  P,  42°  59'  N.  In  this  case,  the 
two  latitudes  are  so  nearly  equal  that  we  might  use  either 
of  them  as  an  argument  in  Table  2  without  material  inaccu- 
racy. In  fact,  in  using  Table  2  it  is  unnecessary  to  consider 
minutes  of  latitude,  the  nearest  degree  being  sufficient. 

But  often  the  two  latitudes  available  at  this  stage  of  the 
problem  differ  by  many  degrees.  In  such  cases  mariners 
always  use  the  average  of  the  two  latitudes,  and  call  it  the 
''middle  latitude."  In  the  present  case,  the  middle  latitude 
would  be  found  thus  : 

Initial  latitude  =  42°  11' 

Final  latitude  =  42°  59' 

Sum  =  85°  10' 

^  sum  =  middle  latitude  =  42°  35' 


DEAD  RECKONING  WITHOUT  LOGARITHMS       17 

The  nearest  even  degree  to  42°  35'  is  43°,  and  the  prob- 
lem would  therefore  be  worked  with  the  43°  column  of  middle 
latitude  in  Table  2. 

Before  completing  our  problem  it  is  necessary  to  point 
out  that  while  Table  2  is  intended  primarily  for  changing 
longitude  differences  in  minutes  into  departures  in  miles,  it 
can  also  be  used  (as  stated  at  the  foot  of  each  page)  for  the 
inverse  transformation  of  departures  into  longitude  dif- 
ferences ;  and  this  is  the  transformation  we  must  make  in 
our  present  problem.  It  is  merely  necessary  to  use  the 
departure  (40.5)  in  the  left-hand  column,  at  the  head  of 
which  are  the  words  '^Long.  Diff.  or  Dep.,"  indicating  that 
either  of  these  two  may  be  used  as  the  argument  in  that 
column.  Then,  in  the  43°  column  of  middle  latitude,  we 
find  (using  interpolation)  the  tabular  number  10.8. 

This  means  that  a  longitude  difference  of  40'. 5  corre- 
sponds to  a  departure  of  40.5  —  10.8  miles,  or  29.7  miles. 

But  when  the  table,  as  in  the  present  case,  is  used  for  the 
inverse  transformation,  the  tabular  number  10.8  must, 
before  use,  be  multiplied  by  the  factor  given  at  the  bottom 
of  the  column.  For  the  middle  latitude  43°  this  factor  is 
1 .37 ;  and  so  the  right  tabular  number  becomes,  in  the 
present  case : 

10.8  X  1.37  =  14.8; 

and  as  the  longitude  difference  is  always  greater  than  the 
departure,  it  follows  that  the  departure  of  40.5  miles  gives 
a  longitude  difference  of : 

40.5  +  14.8  =  55'.3  =  0°  55', 

if  we  omit  the  odd  tenths. 

The  initial  longitude  of  the  ship  at  the  point  Y  was 
59°  28'  W.  As  her  40°  course  has  carried  her  nearer  to  Green- 
wich, it  follows  that  her  final  longitude  at  the  point  P  is : 

59°  28'  W.  -  0°  55'  =  58°  33'  W. 

We  shall  now  discuss  the  following  similar  problem : 

A  ship  takes  her  departure  from  a  point  about  one  mile 


18  NAVIGATION 

east  of  Navesink  Highlands  Light,  New  Jersey,  in  the  initial 
lat.  40°  24'  N.,  initial  long.  73°  58'  W.,  and  travels  1377 
miles  on  a  course  of  166°.  What  final  latitude  and  longitude 
does  she  attain  ? 

Entering  the  traverse  table  in  the  column  headed  166°, 
which  is  the  same  as  the  14°  column,  we  find : 

For  dist.  900,  lat.,    873.2,  dep.,  217.7 

For  dist.  400,  lat.,    388.1,  dep.,    96.7 

For  dist.  77,  lat.,      74.7,  dep.,    18.6 

Sums,  1377,          1336.0,            333.0 

To  make  the  large  given  distance  (1377  miles)  come  within 
the  range  of  Table  1,  it  has  been  necessary  to  enter  the  166° 
column  three  times,  with  the  arguments  900,  400,  and  77, 
and  then  to  sum  the  corresponding  tabular  numbers. 

The  latitude  difference,  1336  miles,  is  equivalent  to  1336', 
or  22°  16',  counting,  as  usual,  60'  to  1°.  Then,  since  the 
direction  of  her  course  (166°)  carried  the  ship  to  the  south 
of  her  initial  position  (cf.  Fig.  5,  p.  14,  and  p.  19),  we  have : 

Initial  lat.,  40°  24'  N. 
Lat.  diff.,  22°  16'  N. 
Final  lat.,  18°  8'  N. 
Middle  lat.,  29°  16'  N. 

Now  turning  to  Table  2,  in  the  proper  column  for  middle 
latitude  29° : 

For  dep.  300  tabular  number  is  37.6 
For  dep.  33  tabular  number  is  4.1 
Sums        333  41.7 

As  in  the  former  example,  this  41.7  must  be  multiplied 
by  the  factor  at  the  bottom  of  the  column.  This  factor  is 
1.14.  Multiplying,  we  have:  41.7  X  1.14  =  47.5.  Conse- 
quently, long.  diff.  =  333  +  47.5  =  380'.5  =  6°  20'.5.  Since 
the  direction  of  her  course  (166°)  carried  the  ship  eastward, 
and  therefore  nearer  to  Greenwich,  it  follows  that  her  final 
longitude  is  73°  58'  W.  -  6°  20',  or  67°  38'  W.  The  final 
position  is  therefore :  lat.  18°  8'  N. ;  long.  67°  38'  W. 


DEAD  RECKONING  WITHOUT  LOGARITHMS       19 

The  point  indicated  by  this  final  latitude  and  longitude 
is  just  off  the  entrance  to  the  Mona  Passage,  between  Haiti 
and  Porto  Rico ;  the  given  course  and  distance  would  there- 
fore be  correct  for  a  voyage  from  New  York  to  Mona  Passage 

Additional  similar  problems  are  : 

1.  Initiallat.,40°28'N.;  initial  long.,  73°  50' W.";  course, 
119°;  dist.,  2924  miles.  This  would  take  the  ship  from 
Sandy  Hook  to  St.  Vincent,  Cape  Verde  Islands. 

Ans.  Final  lat.,  16°  50'  N. ;  final  long.,  25°  7'  W. 

2.  Initial  lat.,  40°  10'  N. ;  initial  long.,  70°  0'  W. ;  course, 
75°;  dist.,  2606  miles.  This  would  take  the  ship  from  Nan- 
tucket Lightship  to  Fastnet,  the  nearest  point  of  the  Irish 

coast. 

Ans.  Final  lat.,  51°  24'  N. ;  final  long.,  9°  37'  W. 

Before  proceeding  to  our  second  fundamental  problem 
(p.  8),  it  will  be  well  to  explain  briefly  two  further  points 
of  interest.  The  first  of  these  relates  to  the  method  of  desig- 
nating a  ship's  course.  We  have  hitherto  supposed  it  to 
be  measured  in  degrees,  from  the  north,  around  by  way  of 
the  east,  through  the  south  and  west,  and  so  back  to  the 
north  again.  This  is  the  best  way  to  count  courses,  and  is 
the  way  now  in  use  in  the  United  States  Navy.  Since  a 
whole  circle  contains  360°,  it  follows  that  courses  may  con- 
tain any  number  of  degrees  from  0°  to  360°. 

But  there  is  another  quite  convenient,  although  older,  way 
of  designating  courses,  in  which  a  60°  course,  for  instance,  is 
written  N.  60°  E.,  showing  that  the  ship  must  be  steered  60° 
east  of  north.  In  a  similar  way,  a  120°  course  is  written 
S.  60°  E.,  showing  that  the  helmsman  should  head  her  60° 
east  of  south,  which  would  be  the  same  as  30°  south  of  east, 
or  120°  from  the  north  toward  the  south  by  way  of  east. 

The  second  further  point  of  interest  has  to  do  with  the 
relation  between  Tables  1  and  2.  It  is  possible  to  avoid 
entirely  the  use  of  Table  2,  and  to  transform  longitude  differ- 
ences into  departures,  and  vice  versa,  by  means  of  Table  1 


20  NAVIGATION 

alone.  It  so  happens  that  the  relation  between  these  two, 
for  any  given  middle  latitude,  as,  for  instance,  29°,  is  iden- 
tical with  the  relation  between  distance  and  latitude  difference 
in  Table  1  for  the  course  29°.  In  other  words,  if  we  have 
given  a  middle  latitude  and  a  longitude  difference,  and  wish 
to  find  the  departure,  we : 

Call  the  middle  latitude  a  course,  and 
Call  the  longitude  difference  a  distance ; 

Then,  corresponding  to  that  course  and  distance,  find  from 
Table  1  the  tabular  latitude  difference,  and  it  will  be 
the  required  departure.  The  same  process  can  also  be 
reversed,  so  as  to  find  the  longitude  difference  from  the 
departure. 

While  this  method  with  Table  1  is  quite  correct,  we  believe 
beginners  (at  least)  will  find  the  use  of  Table  2  advantageous 
in  the  solution  of  these  problems,  especially  when  the  middle 
latitude  is  not  very  great. 

Coming  now  to  our  second  fundamental  problem  of  dead 
reckoning,  let  us  suppose  a  ship  is  required  to  proceed  from 
the  initial  lat.  42°  11'  N.  and  long.  59°  28'  W.  to  a  final 
lat.  42°  59'  N.  and  long.  58°  33'  W.  We  are  to  find  the  course 
she  must  steer,  and  the  distance  she  must  run. 

We  have  at  once  the  latitude  difference  of  0°  48',  or  48  miles, 
and  the  middle  latitude  42°  35',  or  nearest  whole  degree  of  mid- 
dle latitude,  43°.  The  longitude  difference  is  55' ;  and  with  this 
we  find  from  Table  2  the  correction  14.8  in  the  43°  column 
of  middle  latitude.  Remembering  that  this  time  we  are 
transforming  a  longitude  difference  into  departure,  and  con- 
sequently do  not  need  to  use  the  factor  at  the  foot  of  the 
column,  we  subtract  this  correction  (14.8)  from  the  longi- 
tude difference  (55')  and  obtain  the  departure  as  40.2  miles. 

Next  we  proceed  to  Table  1,  to  find  the  course  and  distance 
corresponding  to  lat.  48,  dep.  40.2.  To  do  this,  we  must 
find  a  place  in  Table  1  where  this  particular  latitude  and 
departure  appear  side  by  side.     If  this  pair  of  numbers 


DEAD   RECKONING  WITHOUT   LOGARITHMS       21 

cannot  be  found  (exactly)  side  by  side,  we  must  take  the 
pair  which  come  nearest  to  them :  in  this  case  such  a  pair 
of  numbers  is  found  in  the  40°  course  column,  opposite  dist. 
63.  So  it  appears  that  the  ship  must  steer  on  a  40°  course 
a  distance  of  63  miles,  to  proceed  from  the  given  initial  to 
the  given  final  latitude  and  longitude.  This  problem  is  the 
direct  converse  of  the  one  first  solved  (pp.  15,  17). 

As  a  second  example,  let  us  now  calculate  the  course  and 
distance  from  Sandy  Hook,  lat.  40°  28'  N. ;  long.  73°  50'  W., 
to  St.  Vincent,  lat.  16°  50'  N. ;  long.  25°  7'  W.  We  have, 
by  subtraction,  lat.  diff.  =  23°  38'  =  1418'  =  1418  miles; 
long.  diff.  =  48°  43'  =  2923'. 

This  2923'  must  be  turned  into  a  departure,  the  middle 
latitude  being  28°  39',  or,  to  the  nearest  whole  degree,  29°. 
Turning  to  the  column  of  Table  2  which  belongs  to  29°  of 
middle  latitude,  we  find  the  correction  for  2923'  of  longitude 
difference  thus : 

Tabular  number  for  900  =  113.0, 

which  being  multiplied  by  3,  gives  : 

Tabular  number  for  2700  =  339.0 

Also,  tabular  number  for       200=    25.1 

Tabular  number  for  23  =      2.9 

Sums,  tabular  number  for  2923  =  367.0 

This  must  be  subtracted  from  the  longitude  difference,  and 

so  we  get : 

dep.  =  2923  -  367.0  =  2556  miles. 

We  have  now  to  seek  a  place  in  Table  1  where  lat.  1418  and 
dep.  2556  appear  side  by  side.  No  traverse  tables  are  sufii- 
ciently  extended  to  contain  these  large  numbers,  but  we 
can  at  once  obtain  an  approximate  answer  to  the  problem 
by  dividing  both  numbers  by  100.  This  reduces  them  to 
lat.  14.2,  dep.  25.6  ;  and  the  nearest  numbers  to  these  which 
can  be  found  side  by  side  in  Table  1  are  in  the  column  belong- 
ing to  course  119°  and  opposite  dist.  29.  This  course  (119°) 
is  the  same  as  would  have  been  obtained  if  we  had  not  been 


22  NAVIGATION 

forced  to  divide  our  latitude  and  departure  by  100,  to  bring 
them  within  the  range  of  Table  1.  But  the  dist.  29  must 
now  be  multiplied  by  100,  to  remove  the  effect  of  our  former 
division  of  latitude  and  departure  by  100.  Thus  we  have 
the  closely  approximate  information  that  the  course  and 
distance  from  Sandy  Hook  to  St.  Vincent  are  119°  and  2900 
miles.  The  same  problem  (p.  19),  when  taken  in  its  inverse 
form,  starts  with  the  numbers  119°  and  2924  miles. 

In  discussing  such  a  problem,  many  beginners  have  dif- 
ficulty in  choosing  correctly  the  course  number  (119°)  from 
the  four  (61°,  119°,  241°,  299°)  to  be  found  at  the  foot  of 
the  same  column  of  Table  1.  This  choice  is  easily  made  with 
the  help  of  our  knowledge  of  elementary  geography,  or  with 
any  rough  chart  or  map.  From  these,  we  know  that  St. 
Vincent  is  south  and  east  of  Sandy  Hook,  and  the  only  one 
of  the  four  possible  courses  that  will  carry  a  ship  south  and 
east  is  course  119°.  The  same  course  might  be  written  in 
the  other  notation  (p.  19)  S.  61°  E.,  which  possibly  makes 
the  actual  direction  to  be  steered  a  little  easier  to  under- 
stand. 

The  above  result  is  approximate  only,  but  higher  accuracy 
is  seldom  required.  When  desired,  it  can  be  obtained  by 
certain  kinds  of  interpolations  (p.  12) ;  but  these  are  always 
unsatisfactory,  especially  as  complete  precision  can  always 
be  easily  had  by  the  use  of  logarithms,  as  explained  in  the 
next  chapter. 


CHAPTER  III 
DEAD  RECKONING  WITH   LOGARITHMS 

Since  the  publication  in  1876  of  Kelvin's  tables  for 
facilitating  Sumner's  method,  it  has  been  possible  to  navi- 
gate in  the  most  approved  way  without  using  logarithms  or 
trigonometry.  Those  who  desire  to  study  the  subject  in 
this  manner  may  do  so  by  simply  omitting  those  parts  of 
the  book  in  which  logarithmic  or  trigonometric  formulas 
and  calculations  occur.  But  this  method  of  study  is  not 
recommended,  except  perhaps  for  a  first  reading;  for  a 
knowledge  of  logarithmic  processes  always  affords  a  most 
desirable  check  on  the  accuracy  of  the  other  method,  and 
so  makes  for  safety  of  the  ship  and  peace  of  mind  of  the 
navigator. 

Proceeding,  then,  with  the  subject  of  logarithms,  we  may 
define  them  as  a  mathematical  device  for  facilitating  calcula- 
tions. They  are  merely  numbers ;  but  they  are  numbers 
having  this  peculiarity :  every  logarithmic  number  belongs 
to  some  ordinary  number  (like  1,  2,  3,  27,  800,  etc.),  and 
belongs  to  it  alone.  Its  logarithm  belongs  to  the  number  as 
a  man's  shadow  belongs  to  the  man. 

For  our  present  purpose  it  is  unnecessary  to  enter  into  the 
theory  of  logarithms ;  we  shall  explain  only  the  methods  of 
using  them  in  practice.  Logarithms  (abbreviated  ''log") 
always  consist  of  two  parts,  a  ''whole  number"  part  and  a 
"decimal"  part.  Thus,  3.30103  is  a  logarithm,  of  which 
the  whole  number  part  is  3,  and  the  decimal  part  .30103. 
The  whole  number  part  may  even  be  zero :  thus,  0.30103 
is  also  a  logarithm.  The  decimal  part  of  the  logarithm 
is  found  from  a  table  of  logarithms,  such  as  our  Table  3 

23 


24  NAVIGATION 

(p.  178) ;  but  the  whole  number  part  is  found  by  an  inspec- 
tion of  the  number  to  which  the  logarithm  belongs. 

We  shall  hereafter,  to  save  space,  always  write  ''log  26'^ 
in  place  of  ''the  logarithm  belonging  to  26":  and,  with 
the  help  of  this  abbreviation,  we  may  now  write  the  follow- 
ing tabular  statement,  which  is  fundamental  in  the  matter 
of  logarithms : 

log  1  =  0.00000,  log  1000  =  3.00000, 
log  10  =  1.00000,  log  10000  =  4.00000, 
log  100  =  2.00000,    log  100000  =  5.00000,  etc. 

In  other  words,  for  these  particular  numbers,  all  "mul- 
tiples" of  10,  the  decimal  part  of  the  log  is  zero.  For 
numbers  intermediate  between  1  and  10,  the  whole  number 
part  of  the  log  is  0,  and  the  decimal  part  lies  between 
.00000  and  .99999.  For  those  between  10  and  100  the  whole 
number  part  is  1,  and  the  decimal  part  again  lies  between 
.00000  and  .99999. 

The  general  rule  is :  the  whole  number  part  of  a  log  is 
one  less  than  the  number  of  figures  or  "digits"  in  the  number 
to  which  the  log  belongs.  Thus,  the  number  26  has  two 
digits  :  the  whole  number  part  of  its  log  is  1 .  The  number 
2678  has  four  digits :  the  whole  number  part  of  its  log  is 
therefore  3. 

If  a  number  is  itself  partly  decimal,  we  count  only  the 
number  of  digits  to  the  left  of  the  decimal  point  for  the  pur- 
poses of  the  present  rule.  Thus,  26.78  has  two  digits  only ; 
2.678  has  one ;    267.8  has  three,  etc. 

If,  on  the  other  hand,  a  number  is  wholly  decimal,  as 
0.2678,  the  whole  number  part  of  its  logarithm  should  be 
"negative,"  or  minus,  i.e.  less  than  0;  and  it  will  be  one 
greater  than  the  number  of  zeros  immediately  following  the 
decimal  point  in  the  number.  According  to  this,  the  whole 
number  part  of  log  0.2678  should  be  —  1,  because  this 
number  has  no  zeros  immediately  following  the  decimal 
point.  But  as  these  negative  whole  number  parts  are 
very  inconvenient  in  actual  work,  it  is  customary  to  increase 


DEAD  RECKONING  WITH  LOGARITHMS  25 

all  logs  of  decimal  numbers  arbitrarily  bj^  10,  which  will 
avoid  the  negative  sign.  This  arbitrary  increase  is  always 
corrected  again  in  the  further  or  final  procedure,  so  that  it 
cannot  possibly  introduce  error  into  the  work. 

In  the  case  of  log  0.2678,  the  arbitrary  increase  of  10 
changes  the  —  1  to  +  9  ^ ;  and  so  9  would  be  the  whole 
number  part  of  log  0.2678.  Similarly,  log  0.002678  would 
have  7  for  its  whole  number  part,  because  there  are  two  zeros 
after  the  decimal  point.  This  would  make  the  whole  number 
part  of  the  log   —  3,  which,  being  increased  by  10,  gives  +  7. 

In  general,  this  matter  of  logs  of  wholly  decimal  numbers 
may  be  summarized  as  follows  : 

log  0.1  =9.00000,  log  0.0001  =6.00000, 
log  0.01  =8.00000,  log  0.00001  =5.00000, 
log  0.001  =  7.00000,    log  0.000001  =  4.00000,  etc. 

In  all  these  cases  the  decimal  part  of  the  log  is  zero : 
and  if  the  number  lies,  for  instance,  between  0.1  and  0.01, 
the  whole  number  part  of  the  log  will  be  8,  and  the  decimal 
part  will  lie  between  .00000  and  .99999. 

The  decimal  part  in  the  log  of  any  number  is  taken  from 
Table  3  without  regard  to  the  position  of  the  decimal 
point  in  the  number  itself.  The  numbers  0.2678,  0.002678, 
26.78,  2.678,  267.8,  and  2678  all  have  precisely  the  same 
decimal  part  in  their  logs,  so  that  such  logs  will  differ  in 
their  whole  number  parts  only.  We  can  at  once  obtain  this 
common  decimal  part  from  Table  3  (p.  181),  where  it  is 
found  to  be  .42781.  In  looking  up  this  log,  we  again  use 
(p.  11)  a  pair  of  arguments.  The  argument  for  the. left- 
hand  column  consists  of  the  first  three  digits  of  2678  (267)  ; 
and  in  selecting  this  argument  we  disregard  any  zeros  that 
may  immediately  follow  the  decimal  point,  if  the  number 
is  wholly  decimal,  like  .002678.  The  other  argument,  in 
the  top  horizontal  line  of  the  tabular  page  is  8,  the  right- 
hand   digit   of  the   number   2678.     In  the  horizontal  line 

1  According  to  Algebra,  9  is  greater  than  -  1  by  10. 


26  NAVIGATION 

opposite  267,  and  in  the  column  headed  8,  appears  781 ;  and 
these  are  the  last  three  digits  of  the  required  log  (.42781). 
The  first  two  digits  (.42)  are  common  to  a  great  many  logs, 
and  are  therefore  only  printed  in  the  column  headed  0. 
The  first  two  digits  of  every  log  are  thus  taken  from  the 
zero  column,  regularly  from  the  same  horizontal  line  that 
contains  the  last  three  digits  of  the  log,  or  from  some  line 
above  it.  Only  when  there  is  an  asterisk  printed  in  the  table 
with  the  last  three  digits  do  we  make  an  exception,  and  take 
the  first  two  digits  from  the  line  helow  the  one  containing  the 
last  three.  Thus  the  decimal  part  of  log  2691  is  .42991,  but 
the  decimal  part  of  log  2692  is  .43008. 

Having  thus  found  the  decimal  part  of  log  2678  to  be 
.42781,  and  the  number  2678  having  four  digits,  the  com- 
plete 

log  2678  =  3.42781 ; 

and  here  the  reader  should  once  more  note  that  all  tabular 
logs  like  .42781  are  thus  always  decimals.  The  correspond- 
ing logs  for  the  other  numbers  given  above  are : 

log  267.8  =  2.42781, 
log  26.78  =  1.42781, 
log  2.678  =  0.42781, 
log  0.2678  =  9.42781, 
log  0.002678  =  7.42781. 

It  is  clear  that  Table  3  gives  directly  the  decimal  part  of 

the  logs  of  all  numbers  containing  four  digits.     If  the  number 

contains  less  than  four  digits,  as  26,  we  should  look  it  up  in 

the  table  as  if  it  were  2600.     We  should  find  260  as  the 

argument  in  the  left-hand  column    (p.   181) ;    and   in  the 

corresponding  line,  in  the  column  headed  0  (the  fourth  digit 

of  2600),  is  41497.     This  is  the  decimal  part,  as  usual,  and 

the  complete 

log  26  =  1.41497. 

If,  on  the  other  hand,  the  number  whose  log  is  wanted 
contains  more  than  four  digits,  as  26782,  it  is  necessary  to 


DEAD  RECKONING  WITH  LOGARITHMS     27 

resort  to  interpolation  (p.  12).  The  number  of  digits  being 
here  5,  the  whole  number  part  of  the  log  is  4  (p.  24).  The 
decimal  part  of  the  log  is  to  be  found  quite  without  regard 
to  decimal  points  (p.  25).  It  may  therefore  be  taken 
from  Table  3  just  as  if  we  wanted  log  2678.2  instead  of  26782. 
Now  the  table  tells  us  (p.  181) : 

decimal  part  of  log  2678  =  42781, 
decimal  part  of  log  2679  =  42797. 

The  tabular  difference  (p.  12)  of  these  two  decimal  parts 
is  16.  As  26782  may,  for  our  present  purpose,  be  regarded 
as  lying  y%  of  the  way  from  2678  to  2679,  it  follows  that  the 
decimal  part  of  log  26782  will  lie  ^  of  the  way  from  42781 
to  42797.  Evidently,  we  must  multiply  the  tabular  differ- 
ence 16  by  -^  (giving  3.2)  to  find  how  much  larger  the  decimal 
part  of  log  26782  is  than  the  decimal  part  of  log  2678. 
This  3.2  (or  3,  in  round  numbers)  must  then  be  added  to 
42781 ;  and  we  have,  as  the  result  of  this  interpolation : 

decimal  part  of  log  26782  =  .42784. 

As  we  have  just  found  the  whole  number  part  to  be  4, 
we  have  for  the  complete : 

log  26782  =  4.42784. 

This  whole  process  of  interpolation  may  perhaps  be  more 
clearly  understood  if  we  repeat  (p.  10)  that  all  tables  furnish 
tabular  numbers  corresponding  to  given  arguments.  In- 
terpolation is  necessary  when  the  given  arguments  are  not 
to  be  found  in  the  argument  part  of  the  table,  but  fall 
between  two  of  the  tabular  arguments.  Then  we  obtain 
by  subtraction  the  difference  between  the  given  argument 
and  the  nearest  smaller  argument  contained  in  the  table. 
This  difference  is  the  ''argument  difference"  (abbreviated, 
arg.  diff.),  and  it  should  be  expressed  as  a  decimal  fraction 
of  the  interval  between  two  successive  arguments  (cf.  -^, 
above).  The  tabular  difference  (tab.  diff.)  between  two 
successive  tabular  numbers  being  also  obtained  by  subtrac- 


28  NAVIGATION 

tion,  we  have  only  to  multiply  the  tabular  difference  by  the 
argument  difference  to  find  the  ''interpolation  difference" 
(int.  diff.)-  This  is  then  added  ^  to  the  proper  tabular 
number  (belonging  to  the  above-mentioned  nearest  argu- 
ment given  in  the  table)  to  obtain  the  tabular  number  re- 
quired. 

The  multiplication  of  the  tabular  difference  bj^  the  argu- 
ment difference  is  facilitated  by  certain  little  auxiliary  mul- 
tiplication tables  (called  tables  of  ''proportional  parts") 
printed  in  the  margins  of  many  mathematical  tables.  In 
the  example  given  above,  the  tabular  difference  was  16 ;  and 
Table  3  contains  on  the  proper  page  (p.  181)  a  proportional 
part  table  headed  with  this  same  number  16 ;  and  it  shows 
that  for  an  argument  difference  .2,  and  tabular  difference  16, 
the  interpolation  difference  is  3.2,  just  as  we  found  above. 

Other  examples  of  logarithms  are  : 

log         427  =  2.63043,  log    42765  =  4.63109, 

log       4276  =  3.63104,  log  282374  =  5.45082, 

log    0.4276  =  9.63104,  log            2  =  0.30103, 

log  0.42765  =  9.63109,  log     .0027  =  7.43136. 

The  above  considerations  are  preparatory  only  to  the 
actual  use  of  Table  3  ;  and  they  are  not  yet  quite  complete. 
For  it  is  still  necessary  to  explain  the  inverse  use  (p.  12)  of 
the  table,  or,  in  other  words,  the  finding  of  the  number  to 
which  a  given  log  belongs.  Thus,  if  the  given  log  were 
3.42781,  we  should  begin  by  looking  up  its  decimal  part 
among  the  logs  in  the  table.  Finding  it  there,  we  take  out 
the  number  to  which  it  belongs,  2678.  We  then  put  in  the 
decimal  point  according  to  the  whole  number  part  of  the  log. 
This  being  3,  we  know  (p.  24)  that  the  number  required  must 
contain  4  digits.     Therefore  : 

number  to  which  the  log  3.42781  belongs  =  2678. 

1  Except  when  a  glance  at  the  table  shows  that  the  tabular  num- 
bers are  growing  smaller,  in  which  case  the  interpolation  difference 
must  be  subtracted.  This  never  occurs  in  Table  3,  but  happens  fre- 
quently in  Table  4. 


DEAD   RECKONING   WITH   LOGARITHMS  29 

If  the  given  log  had  been  2.42781,  the  table  would  furnish 
the  same  number  2678,  but  the  decimal  point  would  be 
differently  located.  Because  the  whole  number  part  of  the 
given  log  is  now  2,  we  know  that  the  number  to  which  it 
belongs  has  three  digits,  and  so : 

number  to  which  the  log  2.42781  belongs  =  267.8. 

When  the  given  log  is  not  to  be  found  in  the  table  exactly, 
a  process  of  inverse  interpolation  is,  of  course,  necessary. 
Thus,  if  the  given  log  is  4.42784,  we  look  for  its  decimal 
part  in  the  table,  and  find  it  lies  between 

42781,  which  belongs  to  the  number  2678,  and 
42797,  which  belongs  to  the  number  2679. 

The  decimal  part  of  the  given  log  being  42784  is  greater  by 
3  than  the  nearest  tabular  number  42781.  This  3  is  there- 
fore the  interpolation  difference.  The  tabular  difference  is 
16,  obtained  by  subtraction  between  42781  and  42797.  We 
now  divide  the  interpolation  difference  by  the  tabular  dif- 
ference, which  gives  .2  (^  =  0.2,  in  round  numbers).  This 
.2  is  the  argument  difference,  and  therefore  the  complete 
number  belonging  to  the  decimal  part  of  the  log  (42784) 
is  26782.  The  whole  number  part  of  the  given  log 
being  4,  the  required  number  must  have  5  digits,  and  will 
therefore  be  26782.  Had  the  given  log  been  2.42784,  we 
should  have  arrived  at  the  number  26782  in  just  the  same 
way ;  but  we  should  locate  the  decimal  point  differently. 
The  whole  number  part  of  the  log  being  now  2,  there  should 
be  only  3  digits  in  the  number,  and  we  should  have : 
number  to  which  the  log  2.42784  belongs  =  267.82. 

Other  similar  examples  are : 

log  =  2.71828,  corresponding  number  =  522.73, 
log  =  4.26323,  corresponding  number  =  18333, 
log  =  9.26323,  corresponding  number  =  0.18333, 
log  =  0.21000,  corresponding  number  =  1.6218. 

The  reader  will  perceive,  from  a  consideration  of  these 
interpolated  numbers,  that  work  with  logarithms  is  never 


30  NAVIGATION 

exact,  absolutely.  This  is  inherent  in  the  nature  of  our 
log  tables,  which  really  contain  only  the  decimal  parts  of  the 
logs  carried  out  to  five  places  of  decimals.  Further  decimals 
of  course  exist,  but  are  here  omitted,  because  five  places 
always  give  sufficient  accuracy  for  navigation  calculations. 

The  simplest  calculations  which  are  facilitated  by  loga- 
rithms are  the  ordinary  arithmetical  processes  of  multi- 
pHcation  and  division.  These  processes  can  be  turned  into 
addition  and  subtraction  by  the  use  of  the  following 
principle : 

The  log  of  a  product  is  equal  to  the  siun  of  the  logs  of  the 
factors. 

According  to  this  principle,  if  we  wish  to  multiply  a  series 
of  factors,  we  simply  add  their  logs.  The  sum  is  then  a  log 
and  the  number  to  which  this  log  belongs  is  the  product  of  the 
series  of  factors.  Suppose,  for  instance,  we  wish  to  multiply 
the  factors  2,  3,  and  4.  The  product  should  be  24.  Proceed- 
ing with  logs,  we  have  from  Table  3  : 

log  2  =  0.30103, 

log  3  =  0.47712, 

log  4  =  0.60206, 

log  product  =  sum  =  1.38021, 

and  the  number  to  which  the  log.  1.38021  belongs  is,  accord- 
ing to  Table  3,  24.00,  the  correct  product. 

It  is  evident  that  the  use  of  the  log  table  is  here  of  no 
advantage,  because  the  factors  are  very  small :  but  when 
large  numbers  are  to  be  multiplied  the  advantage  is  very 
great. 

Taking  now  a  similar  simple  example  of  division,  let  us 
divide  6  by  3.  In  division,  evidently,  we  must  subtract 
the  log  of  the  divisor  from  the  log  of  the  dividend,  to  obtain 
the  log  of  the  quotient.     We  have 

log  6  =  0.77815, 

log  3  =  0.47712, 

log  I  =  difference  =  0.30103, 


DEAD  RECKONING  WITH  LOGARITHMS  31 

and  the  number  to  which  the  log  0.30103  belongs  is  2.000, 

the  correct  quotient.     Other  examples  are : 

2.426  X  42.78  x  17.26  =  1791 .3, 
6.242  X  87.24  x  62.71  =  34149, 
2802 


1726 


=  1.6234, 


if  =  0.75. 
24 

In  the  last  example,  we  have 

log  18  =  1.25527, 
log  24  =  1.38021. 

The  subtraction  would  lead  to  a  negative  log  because  1.38021 

is  larger  than   1.25527.     Therefore  we  arbitrarily  increase 

1.25527  by  10,  giving  11.25527,  and  then  the  subtraction 

gives 

log  quotient  =  9.87506, 

which  is  the  log  belonging  to  the  number  0.75,  the  correct 

quotient. 

We  come  now  to  the  solution  of  the  two  fundamental 

problems  of  dead  reckoning  (pp.  8,  10)  by  means  of  logs. 

For  this  purpose  we  must  use  our  Table  4,  in  connection  with 

Table  3.     Table  4  is  called  a  trigonometric  log  table  and 

the  tabular  numbers  in  it  are  certain  logs  known  as : 

sine,  abbreviated  sin,  cotangent,  abbreviated  cot, 
cosine,  abbreviated  cos,  secant,  abbreviated  sec, 
tangent,  abbreviated  tan,     cosecant,     abbreviated  esc. 

It  is  not  our  purpose  to  consider  the  theory  of  trigonom- 
etry, but  it  is  necessary  for  the  reader  to  have 
some  understanding  of  its  practical  applica- 
tions.    If  we  have  a  triangle  QPY  (fig.  6),  we 
notice  that  it  is  made  up  of  six  ''parts,"  the 
three  sides  and  the  three  angles.     Now  it  is  a 
fact  that  if  we  know  any  three  of  these  six     y 
parts,  we  can  calculate  the  other  three  parts.    Fig.  6.  —  Trigo- 
provided  one  of  the  known  parts  is  a  side.         ^°°^^  ^' 
Trigonometry  is  the  branch  of  mathematics  which  enables  us 


32  NAVIGATION 

to  do  this,  and  the  triangle  QPY  is  the  very  triangle  which 
occurs  in  the  two  problems  of  dead  reckoning. 

In  trigonometry,  every  angle  has  belonging  to  it  a  sin, 
cos,  etc.,  just  as  every  number  has  its  log.  These  sines, 
etc.,  can  be  taken  out  of  Table  4  by  means  of  a  pair  of  argu- 
ments in  the  usual  way.  The  two  arguments  are  the  number 
of  degrees  and  the  number  of  minutes  in  the  angle  (p.  9). 
The  number  of  degrees  is  found  in  Table  4  at  the  top  or  bottom 
of  the  page,  and  the  number  of  minutes  in  the  right-hand  or 
left-hand  column.  Each  page  (as,  for  instance,  p.  229)  has 
eight  degree  numbers,  four,  33°,  (213°),  (326°),  and  146°  at 
the  top,  and  four,  123°,  (303°),  (236°),  and  56°  at  the  bottom. 
The  proper  sines,  etc.,  for  all  these  degrees  appear  on  the 
same  page  (p.  229).  When  the  degree  number  is  at  the  top 
or  bottom  of  the  left-hand  column  33°,  (213°),  (303°),  and 
123°,  the  minutes  must  be  taken  from  the  left-hand  column. 
But  when  the  number  of  degrees  is  at  the  top  or  bottom  of  the 
right-hand  column  146°,  (326°),  (236°),  and  56°,  the  minutes 
must  come  from  the  right-hand  column.  And  when  the 
number  of  degrees  comes  from  the  top  of  the  page,  we  must 
look  for  the  proper  sine,  etc.,  in  a  column  having  the  word 
sin,  etc.,  at  the  top.  But  when  the  degree  number  comes 
from  the  bottom  of  the  page,  the  sine,  etc.,  will  be  taken 
from  a  column  having  the  word  sin,  etc.,  at  the  bottom. 
Thus  (p.  229) : 

sin  33°  26'  =  sin  146°  34'  =  cos  56°  34'  =  cos  123°  26'  =  9.74113. 

In  this  way,  sines,  tangents,  etc.,  can  be  taken  from 
Table  4.     Examples  are  : 

sin  28°  32'  =  9.67913,  cot  117°  10'  =  9.71028, 
cos  66°  14' =  9.60532,  sec  12°  40' =  0.01070, 
tan  128°  28'  =  0.09991,    esc  111°  11'  =  0.03038. 

These  sines,  etc.,  are  really  all  logs.  When  the  whole  num- 
ber part  is  9,  it  indicates  that  the  log  belongs  to  a  number 
which  is  wholly  decimal  (see  p.  24),  and  that  the  log  has 
been  arbitrarily  increased  by  10. 


DEAD   RECKONING   WITH   LOGARITHMS  33 

Of  course  these  trigonometric  tables  can  also  be  used  in 
the  inverse  manner.  Thus,  to  find  the  angle  corresponding 
to  the  sin  9.28190,  we  turn  to  p.  207,  and  finding  9.28190  in 
the  sin  column,  we  see  that  the  corresponding  angle  is 
either  11°  2',  191°  2',  168°  58',  or  348°  58'.  When  the  sin, 
etc.,  cannot  be  found  in  the  table  exactly,  we  may  always 
take  the  nearest  one :  interpolation  is  never  practically 
necessary  in  using  the  trigonometric  tables  in  navigation. 
Examples  are : 

sec  =  0.17177,  angle  =  47°  40',  227°  40',  132°  20',  or  312°  20', 
tan  =  0.17177,  angle  =  56°  3',  236°  3',  123°  57',  or  303°  57', 
sin  =  9.17177,  angle  =  8°  32',  188°  32',  171°  28',  or  351°  28', 
cos  =  9.17177,  angle  =  81°  28',  261°  28',  98°  32',  or  278°  32', 
CSC  =  0.17177,  angle  =  42°  20',  222°  20',  137°  40',  or  317°  40', 
cot  =  0.17177,  angle  =  33°  57',  213°  57',  146°    3',  or  326°    3'. 

Having  thus  explained  the  use  of  Table  4,  we  shall  now 
apply  it  to  the  two  problems  of  dead  reckoning.  These 
problems  are : 

1.  To  find  latitude  difference  and  departure  from  course 
and  distance ; 

2.  To  find  course  and  distance  from  latitude  difference 
and  departure. 

These  problems  are  solved  by  means  of  the  following 
formulas,  in  which  the  letter  C  represents  the  course  angle : 

,^ .    I  log  lat.  diff.  =  log  dist.  +  cos  C, 
I  log  dep.         =  log  dist.  +  sin  C. 

_.     f  tan  C  =  log  dep.  —  log  lat.  diff., 

[  log  dist.         =  log  dep.  —  sin  C. 

Sometimes  it  is  preferable  to  find  the  distance  from  the 
latitude  difference  instead  of  the  departure.  We  then  use 
the  following  modification  of  formula  (2)  : 

(2')  log  dist.  =  log  lat.  diff.  -  cos  C. 

Let  us  now  solve  with  these  formulas  our  former  problem 
(p.  18),  in  which  a  ship  traveled  1377  miles  on  a  course  of 
166°.     Applying  formula  (1)  above,  we  have: 


34  NAVIGATION 

log  dist.  (1377)  =  3.13893        log  dist.  (1377)  =  3.13893 

cos  C  ( 166°)  =  9.98690        sin  C  ( 166°)  =  9.38368 


sum  =  log  lat.  difif.         =3.12583^      sum  =  log  dep.  =  2.52261  ^ 

corresponding  lat.  diff.  =  1336.1  corresponding  dep.    =333.1 

These  corresponding  latitude  difference  and  departure 
agree  very  closely  with  the  results  already  found  (p.  18) 
from  Table  1. 

If  the  departure  and  latitude  difference   were  given,  we 

could  find  the  course  and  distance  by  means  of  formula  (2)- 

In  the  present  case  we  have : 

log  dep.  (333.1)  =  2.52261  log  dep.  (333.1)  =  2.52261 

log  lat.  diff.  (1336.1)  =  3.12583  sin  C  (166°)  =  9.38368 

by  subtraction,  tan  C  =  9.396782  by  subtraction,  log  dist.  =  3. 13893 ^ 

corresponding  C         =  166°  corresponding  dist.  =  1377 

These  numbers,  166°  and  1377  miles,  are  the  same  numbers 
with  which  we  began  this  calculation ;  so  it  is  clear  that  the 
log  method  of  calculation  agrees  with  the  traverse  table 
method.     For  accuracy  the  log  method  is  superior. 

The  transformations  of  departure  into  longitude  differ- 
ence, and  vice  versa,  are  accomplished  logarithmically  with 
the  following  formulas : 

(3)  log  long.  diff.  =  log  dep.—  cos  middle  lat. 

(4)  log  dep.  =  log  long.  diff.  +  cos  middle  lat. 

Thus  the  longitude  difference  corresponding  to  dep.  333.1 

would  be  calculated  by  formula  (3)  as  follows : 

log  dep.  (333.1)  =2.52261 

cos  mid.  lat.  (29°  16',  p.  18)  =  9.94069 
by  subtraction,  log  long.  diff.  =  2.58192 
corresponding  long.  diff.  =  381 '.9  =  6°  21 '.9. 

1  These  numbers  have  been  diminished  by  10,  to  allow  for  the  fact 
that  both  cos  C  and  sin  C  have  been  arbitrarily  increased  by  10  (p. 
32;  cf.  also  p.  25). 

2  This  number  has  been  increased  by  10,  and  therefore  is  in  accord 
with  the  usual  practice  of  avoiding  negative  whole  numbers  in  the 
trigonometric  Table  4. 

3  This  subtraction  is  correct,  if  we  remember  that  the  9.38368  is 
really  too  large  by  10. 


DEAD  RECKONING  WITH  LOGARITHMS     35 

This  is  in  close  accord  with  the  result  on  p.  18,  where 
Table  2  gave  6°  20'. 5.  The  logarithmic  method  is  again 
the  more  precise,  for  it  takes  account  of  minutes  in  the  course, 
which  were  neglected  on  p.  18.  But  either  result  is  accurate 
enough  for  practical  purposes. 

Before  finally  leaving  these  problems  of  dead  reckoning, 
we  shall  explain  briefly  two  additional  methods  of  solving 
them  which  differ  from  the  method  so  far  employed.  These 
two  additional  methods  are  called  ''Mercator  sailing"  and 
''great  circle  sailing";  whereas,  up  to  the  present,  we  have 
been  using  ''middle  latitude  sailing,"  so  named  because 
the  middle  latitude  appears  in  the  calculations. 

Mercator  sailing  is  based  on  a  kind  of  chart  first  designed 
by  Gerhard  Mercator,  a  sixteenth  century  geographer. 
Such  charts  are  still  widely  used  for  nautical  purposes. 
In  calculations  based  on  them,  every  parallel  of  latitude  is 
referred  directly  to  the  equator  by  means  of  a  table  of  "merid- 
ional parts."  Our  Table  5  is  such  a  table,  and  it  gives  the 
meridional  part  for  every  degree  and  minute  of  latitude 
from  the  equator  to  60°.  These  meridional  parts  are  really 
the  distances  from  the  equator  to  the  several  parallels  of 
latitude,  such  as  they  would  appear  on  a  Mercator  chart 
drawn  to  such  a  scale  that  1'  of  longitude  at  the  equator  would 
occupy  one  linear  unit  on  the  chart.  Thus  the  meridional 
part  for  lat.  40°  is  given  in  Table  5  as  2607.6.  Suppose  the 
scale  of  the  chart  at  the  equator  were  1  inch  to  the  degree  of 
longitude.  That  would  be  -^  inch  to  the  minute.  The  dis- 
tance on  the  chart  from  the  equator  to  the  40°  parallel  of 
latitude  would  then  be  2607.6  X  -^  inches  =  43.46  inches. 
It  is  needless  to  say  that  a  chart  on  such  a  scale  could  not 
show  a  very  large  part  of  the  ocean  on  a  single  sheet. 

Calculations  by  Mercator  sailing  are  of  course  only  made 
when  the  distances  involved  are  large  and  great  accuracy  is 
required.  It  is  therefore  best  to  do  them  by  means  of 
logarithms,  although  it  is  also  possible  to  obtain  Mercator 
results  from  the  traverse  table.    In  such  calculations  we  do  not 


36  NAVIGATION 

use  the  latitude  difference  of  ordinary  middle  latitude  sailing. 
In  its  place  appears  the  '' meridional  latitude  difference"  (ab- 
breviated mer.  lat.  diff.),  defined  as  the  difference  between  the 
meridional  parts  (Table  5)  belonging  to  the  two  latitudes 
(initial  and  final)  involved  in  the  problem.  With  this  defini- 
tion in  mind  we  may  now  give  the  Mercator  formulas  as 
follows : 

(5)  log  mer.  lat.  diff.  =  log  long.  diff.  +  cot  C 

(6)  log  long.  diff.         =  log  mer.  lat.  diff.  +  tan  C. 

(7)  tan  C  =  log  long.  diff.  —  log  mer.  lat.  diff. 

Let  us  now  apply  these  formulas  to  the  problem  of  pp.  18 
and  33,  in  which  a  ship  starts  from  the  initial  lat.  40°  24'  N. ; 
long.  73°  58'  W.,  and  travels  1377  miles  on  a  course,  C, 
of  166°.  What  final  latitude  and  longitude  does  she  at- 
tain? The  latitude  difference  is  found  in  the  ordinary  way 
(p.  34),  there  being  no  special  Mercator  formula  for  it,  and 
comes  out  1336.1  miles,  or  1336M  =  22°  16'.  The  final  lati- 
tude (p.  18)  is  therefore  40°  24'  -  22°  16'  =  18°  8'.  Then, 
from  Table  5,  we  have : 

for  initial  lat.  40°  24',  mer.  parts  =  2638.9 
for  final  lat.  18°  8',  mer.  parts  =  1099.4 
by  subtraction,!  mer.  lat.  diff.       =  1539.5 

Now,  applying  formula  (6),  we  have: 

log  mer.  lat.  diff.  (1539.5)  (Table  3,  p.  179)  =  3.18738 
tan  C  (166°)  (Table  4,  p.  209)  =  9.39677 

by  addition,  log  long.  diff.  =  2.58415 

corresponding  long.  diff.  (Table  3,  p.  183)   =  383'.8  =  6°  24' 

The  final  longitude  is  therefore  73°  58'  -  6°  24'  =  67°  34'  W., 
whereas  we  obtained  before  67°  38'  W.  (p.  18). 

Finally,  we  shall  apply  the  Mercator  method  to  the 
example  of  p.  21.  It  is  required  to  find  the  course  and 
distance  from 

Sandy  Hook,  lat.  40°  28'  N. ;  long.  73°  50'  W.  to 
St.  Vincent,    lat.  16°  50'  N. ;  long.  25°    7'  W. 

1  If  one  latitude  were  in  the  southern  hemisphere  and  the  other 
in  the  northern,  we  should  add  the  meridional  parts. 


DEAD   RECKONING  WITH   LOGARITHMS  37 

We  have  from  Table  5 : 

for  initial  lat.  40°  28',  mer.  parts  =  2644.2 
for  final  lat.  16°  50',  mer.  parts  =  1018.1 
by  subtraction,  mer.  lat.  diff.        =  1626.1 

The  longitude  difference  is  found  by  subtraction  to  be 
73°  50'  -  25°  T  =  48°  43'  =  2923'.  Now  applying  formula 
(7),  we  have : 

log  long.  diff.  (2923)  (Table  3)  =  3.46583 
log  mer.  lat.  diff.  (1626)  (Table  3)=  3.21112 
by  subtraction,  tan  C  =  0.25471 

and  therefore  (Table  4)  C  =  119°  5'. 

The  distance  is  found  in  the  ordinary  way  from  the 
latitude  difference  {not  mer.  lat.  diff.)  by  means  of  formula 
(20,  p.  33. 

The  latitude  difference  is  40°  28'- 16°  50' =  23°  38' =  1418'. 

Formula  (2')  then  gives  : 

log  lat.  diff.  (1418')  (Table  3)  =  3.15168 
cos  C  (119°  5')  (Table  4)  =  9.68671^ 

by  subtraction,  log  dist.  =  3.46497  ^ 

corresponding  dist.  (Table  3)  =2917 

Course  119°  5',  distance  2917  miles  is  therefore  the 
solution  by  Mercator  sailing.  On  p.  22,  we  obtained  119° 
and  2900  miles;  and  on  p.  19  we  began  with  119°  and  2924 
miles.     The  agreement  is  satisfactory. 

Having  thus  briefly  described  Mercator  sailing,  we  come 
next  to  ''great  circle  sailing."  This  is  a  method  of  determin- 
ing the  ship's  course  toward  her  port  of  destination  in  such  a 
way  that  the  distance  to  be  traveled  will  be  as  short  as 
possible.  If  the  earth's  surface  were  flat  instead  of  spherical, 
the  shortest  course  would  be  a  straight  line,  as  used  in  plane 
sailing;  but  on  the  sphere  the  shortest  course  is  a  curve 
called  a  "great  circle."  Evidently,  on  all  long  voyages,  the 
great  circle  course  is  the  most  advantageous  one ;  that 
mariners  do  not  more  frequently  use  it  is  due  to  a  peculiarity 
of  their  charts. 

1  This  log  is  really  too  large  by  10,  so  the  subtraction  is  correct. 


38  NAVIGATION 

We  cannot  here  enter  into  the  details  of  chart  "pro- 
jections," as  the  theory  of  chart  making  is  called.  It  is 
sufficient  to  remark  that  a  straight  line  drawn  on  the  ordi- 
nary nautical  charts  (which  follow  the  Mercator  system), 
between  any  two  ports,  will  not  represent  the  shortest  (or 
great  circle)  course  between  them.  On  such  a  chart,  the 
great  circle  course  between  the  two  ports  will  appear  to  be 
longer  than  the  straight  line  course,  although  it  is  really 
shorter.  This  accounts  for  the  use  of  the  longer  Mercator 
course  by  many  navigators. 

Now  there  is  a  kind  of  chart,  called  a  ''great  circle  sailing" 
chart,  on  which  straight  lines  between  ports  really  represent 
shortest  (or  great  circle)  courses.  One  would  therefore 
naturally  suppose  that  mariners  would  entirely  discontinue 
the  use  of  Mercator  charts  in  favor  of  great  circle  charts. 
But  there  is  a  reason  for  not  doing  this. 

On  Mercator  charts,  all  terrestrial  longitude  meridians 
are  represented  by  parallel  vertical  straight  lines.  Conse- 
quently, if  we  draw  another  straight  line  on  the  Mercator 
chart  joining  two  ports,  that  line  will  make  the  same  course 
angle  (p.  10)  with  all  the  meridians.  In  this  way,  a  navigator 
can  get  from  a  Mercator  chart,  by  simply  drawing  a  straight 
line,  and  quite  without  calculation,  a  course  angle  which  will 
carry  him  from  one  port  to  another.  And  because  the  course 
angle  so  obtained  is  the  same  with  respect  to  all  meridians 
to  be  crossed  by  the  ship  it  follows  that  the  voyage  can  be 
completed  (theoretically  at  least)  from  the  one  port  to  the 
other  with  the  great  advantage  of  never  changing  the  course 
to  be  steered. 

On  the  other  hand,  the  great  circle  track  makes  a  different 
angle  with  every  meridian  it  passes :  so  that  the  mariner 
must  make  very  frequent  changes  in  the  course  angle  to  be 
steered  during  the  progress  of  a  voyage.  The  simple 
Mercator  track,  without  change  of  course,  is  called  a  ''rhumb 
line"  ;  the  serious  objection  to  it  is  that  it  sometimes  leads 
to  greatly  (and  unnecessarily)- lengthened  voyages. 


DEAD   RECKONING  WITH   LOGARITHMS  39 

The  final  conclusion  is  that  Mercator  charts,  on  account  of 
their  simplicity,  are  most  convenient  for  short  voyages,  or 
for  parts  of  long  voyages  when  the  land  is  not  far  away. 
But  for  shaping  the  main  part  of  the  course  on  a  very  long 
voyage,  great  circle  sailing  charts  are  to  be  preferred. 

At  times,  in  order  to  avoid  very  high  latitudes,  or  to  round 
some  projecting  point  of  land,  navigators  must  substitute  for 
a  single  great  circle  track  one  '^ composed"  of  two  or  more 
shorter  arcs  of  great  circles.  This  is  called  '^composite" 
sailing. 

Finally,  for  the  sake  of  completeness,  we  shall  merely 
mention  two  other  kinds  of  sailing.  '' Parallel"  sailing,  which 
is  simply  middle  latitude  sailing  when  the  latitude  difference 
is  zero;  and  ^'traverse"  sailing,  from  which  the  traverse 
table  gets  its  name.  This  is  also  the  same  thing  as  middle 
latitude  sailing;  but  the  special  word  ''traverse"  is  used 
when  the  ship  changes  her  course  frequently,  perhaps  even 
during  a  single  day.  It  is  then  possible  to  sum  up  the 
result  of  all  the  short  courses  which  together  make  up  the 
day's  run.  It  is  merely  necessary  to  take  from  the  traverse 
table  the  latitude  difference  and  departure  for  each  short  course 
separately,  and  then  to  add  ^  all  the  values  of  latitude  differ- 
ence for  a  ''summed  latitude  difference,"  and  all  the  values 
of  departure  for  a  "summed  departure."  With  these  a 
"composite  course  and  distance"  can  be  taken  from  the 
traverse  table,  or  calculated  with  logs,  and  these  will  repre- 
sent the  motion  of  the  ship,  just  as  if  she  had  steered  an 
unchanged  course  during  the  entire  day. 

1  It  is  necessary  to  sum  separately  latitude  differences  represent- 
ing northward  motion  of  the  ship  and  those  representing  southward 
motion.  The  difference  of  the  two  sums  is  what  we  need  to  know. 
The  same  is  true  of  departures  representing  eastward  and  westward 
motion  of  the  ship. 


CHAPTER  IV 
THE  COMPASS 

The  ship's  course  has  been  defined  (p.  8)  as  the  angle 
between  the  north  and  the  direction  in  which  the  ship  is 
saiUng.  To  ascertain  what  this  angle  is,  or,  in  other  words, 
to  steer  the  ship,  mariners  use  the  compass.  The  dial  (or 
''card")  of  this  instrument  is  divided,  like  any  circle,  into 
360°.  In  the  United  States  Navy  these  are  numbered  in 
such  a  way  (fig.  7)  that  0°  appears  at  the  north,  90°  at  the 
east,  180°  at  the  south,  and  270°  at  the  west.  The  numbers 
therefore  increase  in  a  ''clockwise"  direction.  There  are 
also  compasses  in  which  the  numbering  begins  with  0°  at 
both  the  north  and  south  points,  and  increases  to  90°  at  the 
east  and  west  points.  But  the  United  States  Navy  system 
of  numbering  is  to  be  preferred. 

In  addition  to  the  above  division  and  numbering,  the  dial 

is  also  divided  into  32  points  (pp.  10,  15),  each  containing 

360° 

,  or  11t°.     These  points  are  then  further  subdivided 

32    '  *  ^ 

into  quarter  points,  all  of  which  is  shown  clearly  in  Fig.  7. 

The  naming  of  the  points  has  not  been  done  by  chance, 
but  in  accordance  with  a  definite  rule.  The  four  principal, 
or  "cardinal,"  points  are  north,  east,  south,  and  west.  The 
remaining  points  are  located  by  a  continued  process  of 
halving.  Halfway  between  the  cardinal  points  are  the 
"inter-cardinal"  points;  and  each  is  named  by  combining 
the  names  of  the  two  cardinal  points  adjacent  to  it.  Thus 
northeast  (abbreviated  N.E.)  is  halfway  between  north 
and  east.  Again  halving  and  combining  names,  we  get 
points  like  E.N.E.,  S.S.E.,  etc.  Still  once  more  halving 
completes  the  tally  of  32  points :  but  a  combination  of 
names   would   now   be   too   complicated.     However,    since 

40 


THE   COMPASS 


41 


each  of  these  final  points  must  necessarily  be  adjacent  to  a 
cardinal  or  inter-cardinal  point,  they  are  named  by  simply 
increasing  the  name  of  such  adjacent  cardinal  or  inter- 
cardinal  point.     This  is  accomplished  with  the  word  "by." 


Fig.  7.  —  Compass  Card. 


Thus  we  find,  adjacent  to  N.E.,  the  points  N.E.  byE.,  and 
N.E.  by  N.  In  the  light  of  the  above,  it  is  easy  to  ''box" 
the  compass,  as  seamen  say,  or  to  name  the  32  points  in 
order. 

When  the  point  system  of  division  is  used,  and  an  accuracy 


42  NAVIGATION 

closer  than  a  single  point  is  required,  the  compass  card  is 
still  further  subdivided  into  quarter  points.  In  naming 
these  it  is  customary,  in  the  United  States  Navy,  to  '^box" 
from  N.  and  S.  towards  E.  and  W.  Thus  the  space  between 
N.N.E.  and  N.E.  by  N.  would  be  divided  into  four  parts 
thus :  N.N.E.iE.,  N.N.E.JE.,  N.N.E.f E.  But  an  excep- 
tion is  made  to  this  last  rule  in  the  case  of  quarter  points 
adjacent  to  a  cardinal  or  inter-cardinal  point.  These  last 
are  always  put  first  in  naming  the  quarter  points.  Thus, 
between  E.  by  N.  and  E.,  if  we  always  boxed  from  N.  towards 
E.,  we  should  have  :  E.  by  N.JE.,  E.  by  N.iE.,  E.  by  N.f E. 
But  it  is  customary,  because  shorter,  to  name  these  quarter 
points  E.fN.,  E.iN.,  and  E.^N. 

Inside  the  ''bowl"  of  the  compass,  and  adjacent  to  the 
card,  a  black  line  is  marked  on  the  bowl.  This  line  is  in 
plain  view  of  the  steersman,  through  the  glass  cover  of  the 
compass,  and  is  called  the  ''lubber  line."  When  the  ship 
is  headed  in  such  a  way  that  this  line  comes  opposite  N.E., 
for  instance,  on  the  card,  the  ship  will  be  on  a  N.E.  course, 
which  makes  an  angle  of  45°  with  the  north. 

But  would  the  ship  really  be  traveling  on  a  line  making 
a  45°  angle  with  the  geographic  meridian,  or  direction  of 
the  north  pole  of  the  earth?  She  would  be  doing  so  only 
if  the  compass  were  absolutely  correct.  This  is  practically 
the  case  with  the  "gyro-compass,"  a  mechanical  contrivance 
now  much  used  in  the  navy,  but  not  the  case  with  the  ordi- 
nary "magnetic"  compass. 

In  Chapters  II  and  III,  concerning  dead  reckoning,  we  have 
always  used  the  word  "course"  as  if  all  compasses  were 
absolutely  correct.  But  since  they  are  not  correct,  it  is 
now  necessary  to  make  allowance  for  their  errors.  In  other 
words,  whenever  we  use  a  compass,  we  must  first  ascertain 
the  difference  between  the  "true  course"  and  the  "compass 
course."  It  must  not  be  supposed  from  this  statement  that 
a  ship  can  be  steered  on  two  different  courses  at  the  same 
moment.     There  is  really  only  one  direction  along  which 


THE   COMPASS  43 

the  ship  is  moving :  but  the  angle  between  that  direction 
and  the  true  north  may  be  different  from  the  angle  between 
it  and  the  ''compass  north."  It  is  the  course  measured 
from  the  true  north  that  must  be  used  in  all  dead-reckoning 
calculations,  and  that  always  results  from  such  calculations : 
but  for  steering  the  ship  by  means  of  a  compass  the  steers- 
man must  be  furnished  with  the  course  as  measured  from 
the  compass  north.  Therefore  it  is  essential  for  the  navigator 
to  know  the  difference  between  the  two.  This  difference 
is  called  the  ''error"  of  the  compass. 

Unfortunately,  this  error  is  made  up  of  two  parts.  The 
first,  called  "variation"  of  the  compass,  is  due  to  peculiari- 
ties in  the  earth's  magnetism,  and  is  quite  different  in  dif- 
ferent places  on  the  earth.  It  also  varies  in  different  years 
at  the  same  place.  But  at  any  one  time,  all  ships  in  the 
same  part  of  the  ocean  will  have  the  same  variation. 

The  mariner  can  always  ascertain  how  great  the  varia- 
tion is  in  his  part  of  the  ocean,  because  it  is  always  marked 
on  his  chart.  Certain  curved  lines  are  drawn  on  the  chart ; 
and  if  the  ship  is  located  on  or  near  a  line  marked  "varia- 
tion 10°,"  for  instance,  it  follows  that  the  navigator  must 
on  that  day  allow  for  10°  of  variation.  It  is  also  important 
to  take  into  consideration  possible  changes  in  the  variation. 
Sometimes  the  annual  change  is  marked  on  the  chart;  if 
not,  it  is  important  to  use  a  chart  of  recent  date. 

The  second  part  of  the  error  is  called  "deviation"  and  is 
due  to  peculiarities  in  the  magnetism  always  developed  in 
the  metallic  parts  of  the  ship  itself.  It  is  different  in  dif- 
ferent ships,  even  in  the  same  part  of  the  ocean,  and  is  even 
different  in  the  same  ship,  when  she  is  headed  on  different 
courses.  Methods  have  been  invented  for  "compensating" 
marine  compasses,  so  as  to  remove  the  effects  of  deviation, 
and  these  methods  are  quite  effective.  But  even  when 
they  are  used,  it  is  necessary,  before  beginning  a  long  voyage, 
to  have  a  "compass  adjuster"  visit  the  ship.  He  will  then 
"swing"  the  ship  on  a  number  of  different  courses,  and 


44  NAVIGATION 

adjust  the  compass  so  that  it  will  be  as  nearly  correct  as  pos- 
sible. Finally,  he  will  determine,  by  means  of  astronomic  or 
other  observations,  just  what  the  remaining  compass  devia- 
tion is  on  all  the  various  courses,  and  give  the  navigator  a 
table  of  these  remaining  deviations.  This  table  must  be  taken 
into  account  in  '^ shaping"  the  ship's  course  during  the 
voyage.  The  navigator  must  also,  from  time  to  time,  check 
these  tabular  deviations  while  at  sea  by  means  of  astronomic 
observations  of  his  own,  to  take  care  of  possible  changes. 

Such  astronomic  observations  are  made  with  an  instru- 
ment (the  ''azimuth  circle"),  which  can  be  attached  to  the 
compass,  and  with  which  the  ''compass  bearing"  of  the 
sun  or  any  other  object  can  be  observed.  The  compass 
bearing  is  simply  the  compass  direction  of  the  object,  as 
seen  from  the  ship ;  or  the  compass  course  on  which  the  ship 
would  l^e  steered,  if  she  were  moving  directly  toward  the 
object.  When  the  sun  is  used,  its  true  bearing,  measured 
from  the  true  north,  can  be  taken  from  astronomic  tables 
which  will  be  explained  later ;  and  it  is  called  the  sun's 
"azimuth."  A  comparison  of  this  true  bearing  with  that 
measured  on  the  compass  with  the  azimuth  circle  then  makes 
the  compass  error  known. 

When  it  is  not  convenient  to  observe  the  sun,  it  is  possible  to 
substitute  observations  of  a  distant  well-defined  terrestrial  ob- 
ject, whose  true  bearing  can  be  measured  on  a  chart  for  com- 
parison with  various  compass  bearings  observed  while  the  ship 
is  being  swung.  Another  method  is  to  set  up  a  compass  on 
shore,  away  from  any  iron  or  steel,  and  use  it  to  determine 
the  bearing  of  the  distant  object.  And  there  is  still  another 
method,  if  the  above  compass  and  the  ship's  compass  are  inter- 
visible.  For  the  bearing  of  each  may  then  be  taken  from  the 
other,  and  these  should  differ  by  exactly  180°.  If  they  do  not, 
the  variation  from  180°  must  be  due  to  deviation  on  board. 

The  "pelorus"  is  another  instrument  which  may  at  times 
replace  the  azimuth  circle.  It  is  located  anywhere  on  the 
ship,  at  a  convenient  point  for  observation,  and  not  neces- 


THE   COMPASS 


45 


sarily  close  to  the  compass.  It  has  a  ''dummy  card"  and  a 
lubber  line.  The  dummy  card  can  be  turned  until  the 
lubber  line  indicates  the  same  course  as  the  real  compass. 
Observations  of  bearings  with  the  pelorus  will  then  obviously 
be  the  same  as  if  made  on  the  compass  with  the  azimuth  circle. 
The  advantage  of  the  pelorus  is  that  it  can  be  used  anywhere 
on  board,  while  the  compass  must  be  kept  constantly  in  the 
exact  place  where  it  was  ''adjusted"  before  leaving  port. 

The  error  thus  determined  astronomically  or  otherwise 
is  the  sum  of  the  variation  and  deviation.  If  we  indicate 
by  E  the  total  compass  error  in  that  place,  at  that  time,  on 
that  ship,  and  on  that  course ;  by  D  the  deviation  similarly 
described  ;  by  V  the  variation  at  that  time  and  in  that  place  ; 
and  if  all  three  are  counted  from  0°  in  the  usual  direction 
around  the  compass  card,  then 
we  have  the  formula  : 

(1)         E  =  V-i-D. 

By  counting  in  the  usual  direc- 
tion, we  mean  counting  from  the 
north  around  to  the  east,  as  all 
courses  are  counted  (p.  19) ;  so 
that  a  compass  error  of  10°,  for 
instance,  would  mean  that  the 
compass  north  pointed  10°  east 
of  the  true  north,  or  had  a  true 
bearing  of  N.  10°  E.  (p.  19). 
This  is  shown  in  Fig.  8,  which 
also  shows  the  ship's  course, 
counted  in  the  same  way. 

It  is  clear  from  the  figure  that  if  we  now  indicate : 

by  C,  the  ship's  compass  course, 

by  T,  the  ship's  true  course, 

by  E,  the  compass  error, 

we  shall  have  the  formula  : 


Fig.  8.  —  Compass  Error. 


(2) 


T  =  C  +  E. 


46  NAVIGATION 

The  simple  formulas  (1)  and  (2)  enable  the  navigator  to 
make  all  necessary  compass  calculations.  The  following 
are  examples. 

Suppose,  for  instance,  that  the  error  E  has  been  deter- 
mined by  observation,  and  the  variation  V  taken  from  the 
chart.  Formula  (1)  then  makes  it  possible  to  calculate 
the  deviation  D.  For  the  formula  shows  that  E  is  the  sum 
of  V  and  D ;  and  so  D  must  be  the  difference  of  E  and  F, 

^^'  D  =  E  -  V. 

Thus  the  deviation  D  becomes  known,  as  a  check  on  the 
compass  adjuster's  work,  and,  while  this  value  of  D  is  cor- 
rect only  for  the  particular  course  on  which  the  ship  was 
headed  at  the  time  the  observation  was  made,  yet  that 
course  is  the  very  one  for  which  it  is  especially  important 
to  have  correct  information. 

Again,  suppose  dead-reckoning  calculations  show  that  the 
ship  is  to  sail  on  a  40°  course.  These  calculations  always 
furnish  the  true  course  (p.  43)  so  that  T  =  40°.  The 
variation  being  known  from  the  chart,  and  the  deviation 
from  the  adjuster's  table,  we  know  from  (1)  E  =  V  -\-  D. 
Then  from  (2)  we  see  that  C  =  T  —  E,  which  gives  the 
compass  course.  Let  us  suppose  in  the  present  case,  that 
V  was  9°,  D  1° ;  then  E  =  V  -{-  D  =  9°  +  1°  =  10° ;  and 
since  T  =  40°,  C  =  T  -  £'  =  40°  -  10°  =  30° ;  and  the 
helmsman  would  be  directed  to  steer  a  30°  course  by  com- 
pass. 

If,  in  Fig.  8,  the  compass  north  happened  to  be  10°  on  the 
left  side  of  the  true  north,  instead  of  the  right,  the  error  E 
would  be  350°,  instead  of  10°  (see  also  fig.  7,  p.  41).  This 
might  be  made  up  of  a  variation  V  of  349°  and  a  deviation 
D  of  1°,  as  before.  If  the  true  course  is  again  to  be  40°, 
the  compass  course  would  be  40°  —  350°,  according  to  the 
formula  C  =  T  —  E.  This  subtraction  being  impossible, 
we  increase  the  40°  by  a  complete  circumference  of  360°. 
which  is  always  permissible,  and  then  have : 


THE   COMPASS  47 

C  =  360°  +  40°  -  350°  =  50°. 

The  ship  would  be  steered  on  a  compass  course  of  50°. 

An  alternative  way  to  take  care  of  errors,  variations, 
and  deviations  on  the  left  side  of  the  true  north  is  to  mark 
them  with  the  negative  or  minus  sign.  Instead  of  calling 
V  349°,  we  might  call  it  —  11°.  This  is  really  the  best  way, 
and  leads  to  the  same  result  as  before,  if  we  remember  that 
the  subtraction  of  a  minus  quantity  is  always  equivalent  to 
an  addition.  In  the  example  just  given,  calling  V  —  11°, 
instead  of  349°,  we  should  have :  E  =  V  +  D  =  -  11°  -{- 
1°  =  _  10°;  and  C  =  T  -  E  =  40°  -  (-  10°)  =  50°,  the 
same  compass  course  as  before. 

An  older  way  of  designating  variations,  deviations,  and  errors 
is  to  call  them  east  when  the  compass  north  points  to  the 
right  of  the  true  north,  and  west  when  it  points  to  the  left 
of  the  true  north.  This  method  leads  to  the  necessity  of 
providing  various  rules  or  diagrams  with  which  to  make 
compass  calculations.  We  think  the  best  way  to  avoid 
error  (and  such  errors  may  lose  ships  and  lives)  is  to  use  the 
method  here  given  with  its  two  simple  formulas.  When 
some  other  designation  of  the  error,  or  some  other  method 
of  numbering  the  card,  is  demanded  by  a  captain,  it  is  always 
possible  to  conform  to  that  demand,  but  also  to  translate 
every  problem  into  our  method  (in  imagination  at  least) 
as  a  check  against  mistake. 

The  following  is  an  example  of  a  compass  adjuster's  ''devia- 
tion table,"  taken  from  Bowditch's  ''Navigator"  (1916 
edition) .  The  deviations  are  set  down  in  degrees  and  tenths 
of  a  degree,  instead  of  degrees  and  minutes,  for  convenience 
in  the  further  calculations.  The  ship  was  swung  so  that 
her  head  bore  successively  around  the  horizon,  and  obser- 
vations were  made  at  intervals  of  15°.  This  is  a  smaller 
interval  than  is  usually  necessary ;  and  the  deviations  in  the 
table  are  much  larger  than  commonly  occur  in  a  modern 
well-compensated  compass. 


48 


NAVIGATION 


DEVIATION   TABLE 


Bearing 

Bearing 

Bearing 

Bearing 

OF  Ship's 

Devia- 

of Ship's 

Devia- 

of Ship's 

Devia- 

o)? Ship's 

Devia- 

Head  by 

tion 

Head    by 

tion 

Head   by 

tion 

Head    by 

tion 

Compass 

Compass 

Compass 

Compass 

o 

o 

o 

0 

o 

o 

o 

o 

0 

-  15.5 

90 

-  9.1 

180 

+  17.9 

270 

+    9.9 

15 

-  14.9 

105 

-  9.0 

195 

+  23.8 

285 

+    1.9 

30 

-  13.3 

120 

-  7.8 

210 

+  27.1 

300 

-    4.2 

45 

-  11.3 

135 

-  5.9 

225 

+  25.6 

315 

-  10.3 

60 

-  10.0 

150 

-  2.3 

240 

+  22.0 

330 

-  13.6 

75 

-    9.7 

165 

H-8.5 

255 

+  15.9 

345 

-  16.0 

To  illustrate  the  use  of  this  table,  let  us  suppose  the  ship 
to  be  sailing  on  a  compass  course  of  165°,  in  a  part  of  the 
ocean  where  the  variation  is  +  10°,  or  10°  E.  Using  formula 
(1)  (p.  45),  and  finding  from  our  table  that  the  deviation  D 
for  165°  is  +  8°.5,  we  have  the  compass  error  E  =  V  -{-  D 
=  +  10°  +  8°.5  =  +  18°.5.  By  formula  (2)  (p.  45)  the  true 
course  of  the  ship  is  T  =  C -{- E  =  165°  +  18°.5  =  183°.5. 
We  should  use  this  true  course  183°.5  in  calculating  later 
the  ship's  position  by  dead  reckoning  (p.  10). 

If  the  compass  variation  were  everywhere  the  same,  it 
would  be  more  convenient  to  have  a  table  of  compass  errors, 
instead  of  a  table  of  deviations ;  but  because  the  variation, 
as  given  on  the  chart,  varies  greatly,  the  table  must  be 
specially  made  for  deviations  only. 

Equally  important  with  the  above  use  of  our  deviation 
table  is  its  inverse  use.  When  the  navigator  has  calculated 
by  dead  reckoning  the  course  he  must  steer,  that  course, 
as  it  comes  from  the  calculations,  will  be  a  true  course  (p. 
43) ;  and  it  is  necessary  to  turn  it  into  a  compass  course  for 
the  use  of  the  steersman. 

To  do  this  we  must  know  the  deviation ;  and  we  cannot 
get  it  directly  from  the  deviation  table  above,  because  the 
use  of  that  table  presupposes  a  knowledge  of  the  compass 
course,  the  very  thing  we  are  trying  to  find.     The  best 


THE   COMPASS 


49 


way  to  avoid  this  difficulty  is  to  imagine  the  deviation  to  be 
non-existent,  for  the  moment,  and  to  make  use  of  the  ''mag- 
netic course,"  defined  as  the  course  which  would  be  indi- 
cated by  the  compass,  if  deviation  were  thus  totally  absent. 
Under  these  circumstances,  formula  (1)  gives  E  =  V,  since 
D  =  0 ;  and  if  we  designate  the  magnetic  course  by  M,  we 
may  write,  in  place  of  formula  (2)  (p.  45) : 

(3)         M  =  T-V, 

Let  us  suppose  a  case  in  which  the  variation  is  + 10°,  and 
the  desired  true  course  of  the  ship  175°.  Then  the  magnetic 
course,  allowing  for  variation  only,  will  be,  by  formula  (3) : 

M  =  T  -  V  =  175°  -  10°  =  165°. 

This  course  is  not  really  a  compass  course,  because  no 
account  has  yet  been  taken  of  the  deviation.  Nor  can  we 
yet  find  the  deviation  directly  from  the  deviation  table, 
because  in  that  table  we  must  still  know  the  compass  course 
to  use  as  the  argument  (p.  10),  whereas  we  know  as  yet  only 
the  magnetic  course.  Therefore  navigators  should  always 
request  the  compass  adjuster  to  furnish  a  ''second  deviation 
table,"  in  which  the  argument  is  the  magnetic  course,  in- 
stead of  the  compass  course.  Such  a  second  table  can  al- 
ways be  calculated  from  the  other.  We  here  give  one  that 
has  been  calculated  from  the  table  on  the  preceding  page. 


SECOND    DEVIATION    TABLE 


Mag- 

Mag- 

Mag- 

Mag- 

NKTIC 

netic 

netic 

netic 

Bearing 

Devia- 

Bearing 

Devia- 

Bearing 

Devia- 

Bearing 

Devia- 

OF Ship's 

tion 

OF  Ship's 

tion 

OF  Ship's 

tion 

OF  Ship's 

tion 

Head 

Head 

Head 

Head 

o 

0 

o 

o 

o 

0 

o 

o 

0 

-  14.9 

90 

-  9.0 

180 

+  11.0 

270 

+  16.5 

15 

-  13.4 

105 

-  8.4 

195 

+  16.9 

'285 

+    4.1 

30 

-  11.7 

120 

-  6.9 

210 

+  21.3 

300 

-    7.1 

45 

-  10.4 

135 

•  -  4.8 

225 

+  24.9 

315 

-  13.2 

60 

-    9.8 

150 

-  1.4 

240 

+  26.8 

330 

-  15.7 

75 

-    9.3 

165 

+  5.0 

255 

+  24.1 

345 

-  15.5 

E 


50  NAVIGATION 

We  also  add  as  an  example  the  calculation  of  one  number 
in  the  second  table  from  those  given  in  the  first.  We  shall 
find  the  deviation  corresponding  to  the  magnetic  course 
165° ;  and  we  do  it  by  a  kind  of  interpolation  (p.  12).  From 
the  first  table  we  have  the  deviation  —  2°. 3  for  the  compass 
course  150°.  Since  the  deviation  is  the  only  difference 
between  compass  and  magnetic  courses,  it  follows  that 
150°  —  2°. 3,  or  147°. 7  magnetic,  corresponds  to  150°  by  com- 
pass. Similarly,  173°. 5  magnetic  corresponds  to  165°  by 
compass. 

The  magnetic  course  165°  for  which  we  are  making  the 
calculation  falls  between  147°. 7  and  173°. 5,  and  exceeds 
the  smaller  of  the  two  by  17°. 3.  The  whole  difference  be- 
tween 147°.7  and  173°.5  is  25°.8.  Similarly,  the  whole  dif- 
ference between  the  two  compass  courses  involved  is  15°. 
Therefore  we  may  write  the  proportion  : 

25°.8  :  15°  =  17°.3  :  a;°, 

where  x  is  the  excess  over  150°  of  the  compass  course  corre- 
sponding to  165°  magnetic. 

Solving  this  proportion  by  the  ordinary  rules  of  arithmetic, 

we  have : 

15  X  17.3^  jQOQ 

25.8 

The  compass  course  belonging  to  165°  magnetic  is  there- 
fore 150°  H-  10°.0  =  160°. 0.  The  corresponding  deviation 
is  165°  -  160°. 0  =  -j-  5°.0,i  which  is  therefore  the  deviation 
for  165°  magnetic,  and  appears  as  such  in  the  second  table. 
This  entire  table  can  be  computed  from  the  first  table  in  an 
hour. 

Sometimes  the  second  deviation  table  gives  compass  courses 
instead  of  deviations.     It  is  then  often  called  a  ''table  of 

^A  comparison  of  formulas  (1),  (2),  vand  (3)  shows  that 
D=  M  —  C;  so  that  the  deviation  is  obtained  by  subtracting  the 
compass  course  from  the  magnetic  course.  This  is  also  evident 
from  the  definition  of  a  magnetic  course  (p.  49). 


THE   COMPASS  51 

steering  courses " ;  and  in  the  example  just  calculated  it 
would  give  the  compass  or  steering  course  160°  for  the  mag- 
netic course  165°,  instead  of  giving  the  deviation  +5°. 

We  shall  still  further  illustrate  this  important  matter  by 
an  example,  supposed  to  occur  on  board  a  ship  for  which 
our  two  deviation  tables  hold  good. 

What  is  the  compass  course  to  be  given  the  helmsman  at 
Sandy  Hook,  on  a  voyage  to  St.  Vincent? 

We  have  already  found,  from  dead-reckoning  calculations 
(p.  22)  the  course  119°.  Being  the  result  of  a  dead-reckon- 
ing calculation,  this  is  a  true  course.  The  track  chart  of 
the  north  Atlantic  gives  the  variation  at  Sandy  Hook  as 
10°  W.,  or  —  10°.  The  true  course  being  119°,  we  get  the 
magnetic  course,  allowing  for  variation  only,  by  formula  (3), 
M  =  T-V  =  119°  -{-  10°)  =  129°.  The  second  devia- 
tion table  shows  that : 

for  magnetic  course  120°,  the  deviation  is  —  6°. 9,  and 
for  magnetic  course  135°,  the  deviation  is  —  4°. 8. 

Magnetic  course  129°  falls  between  120°  and  135°,  so  that 
an  interpolation  (to  be  extremely  exact)  between  —  6°. 9 
and  —  4°. 8  makes  the  deviation  for  magnetic  course  129° 
come  out  —  5°. 6.     Formulas  (1)  and  (2)  now  give  : 

Error  =E  =  V+D=-  10°-  5°.6  =  -  15°.6 

Compass  course  =  C  =  T-E  =    119°  -(-  15°.6)  =134°.6. 

To  check  this,  we  can  now  solve  the  same  problem  in  the 
inverse  way  with  the  first  deviation  table.  For  the  compass 
course  134°. 6,  this  table  gives  the  deviation  as  —  5°. 9.  The 
variation  being  —  10°,  we  have : 

E  =  V  +  D  =  -10°  -  5°.9  =  -  15°.9  and 
T  =  C  +  E  =  134°.6  -  15°.9  =  118°.7, 

agreeing  very  closely  with  the  true  course  119°,  with  which 
we  started.  This  shows  that  the  two  deviation  tables  are 
quite  consistent  in  this  case,  and  also  checks  the  accuracy 
of  the  calculation. 


52 


NAVIGATION 


We  shall  close  this  chapter  with  the  following  little  table, 
showing  the  correspondence  between  the  two  methods  of 
dividing  the  compass  card  into  points,  and  into  degrees. 

COMPASS  POINTS  AND  DEGREES 


o   / 

o    / 

O      / 

0       / 

North 

0  0 

East 

90  0 

South 

180  0 

West 

270  0 

N.  by  E. 

11  15 

E.  by  S. 

101  15 

S.  by  W. 

191  15 

W.  by  N. 

281  15 

N.N.E. 

22  30 

E.S.E. 

112  30 

s.s.w. 

202  30 

W.N.W. 

292  30 

N.E.  by  N. 

33  45 

S.E.  by  E. 

123  45 

S.W.  by  S. 

213  45 

N.W.  byW. 

303  45 

N.E. 

45  0 

S.E. 

135  0 

S.W. 

225  0 

N.W. 

315  0 

N.E.  by  E. 

56  15 

S.E.  by  S. 

146  15 

S.W.  by  W. 

236  15 

N.W.  by  N. 

326  15 

E.N.E. 

67  30 

S.S.E. 

157  30 

W.S.W. 

247  30 

N.N.W. 

337  30 

E.  by  N. 

78  45 

S.  by  E. 

168  45 

W.  by  S. 

258  45 

N.  by  W. 

348  45 

pt.    -  2°  49' 


i  pt.    =  5°  38' 


pt.    =  8°  26' 


1  pt.   =  11°  15' 


CHAPTER  V 
COASTWISE  NAVIGATION 

Before  proceeding  to  a  consideration  of  navigation  by 
means  of  astronomic  observations,  as  it  is  practiced  on  the 
high  seas,  we  must  first  explain  certain  methods  by  which 
it  is  possible  to  ascertain  a  ship's  position  in  latitude  and 
longitude  while  she  is  in  sight  of  land.  Often  such  methods 
suffice  to  complete  a  long  coastwise  voyage  in  safety ;  they 
are  always  important  for  a  last  determination  of  the  ship's 
position  before  a  deep-sea  voyage  actually  begins.  Such  a 
last  determination  is  called  ''taking  a  departure"  (cf.  p.  2), 
and  from  such  point  of  departure  dead-reckoning  calcula- 
tions begin  for  the  first  day  of  the  voyage. 

Any  determination  or  fixing  of  a  ship's  position,  by  astro- 
nomic observations  or  otherwise,  is  often  called,  for  brevity, 
a  ''fix."  To  obtain  one  while  in  sight  of  land  it  is  customary 
to  make  observations  upon  well-known  objects  ashore, 
such,  for  instance,  as  lighthouses,  or  other  conspicuous 
objects  marked  on  the  chart.  It  is  always  possible  to  ob- 
serve the  bearings  of  such  objects  from  the  ship's  deck  with 
the  compass,  azimuth  circle,  or  pelorus  (p.  44). 

When  there  is  but  one  such  object  in  sight,  it  is  impossible 
to  secure  a  fix  with  ordinary  instruments,  if  the  vessel  is 
at  anchor.  But  if  she  is  running,  it  is  merely  necessary  to 
take  two  bearings,  and  to  estimate  the  distance  run  by  the 
ship  in  the  interval  between  the  two.  Figure  9  will  make 
this  matter  clear.  A  lighthouse  ashore  is  at  L.  SS"  is  the 
direction  of  the  ship's  course ;  S  her  position  when  the 
first  bearing  was  observed,  and  S'  her  position  at  the  time 
of  the  second  bearing.     SN  is  the  direction  of  the  north. 

53 


54 


NAVIGATION 


After  taking  the  first  bearing,  the  navigator  must  calculate 
^the  angle  S''SL,  between  the  ship's  course  SS"  and  the 

lighthouse  direction  SL.  Thus, 
if  the  ship's  course  angle  NSS'^ 
(p.  10)  was  20°,  and  the  bearing 
NSL  was  42°,  the  angle  S^'SL 
would  be  42°  -  20°  =22°.  As 
the  ship  proceeds  on  her  course, 
the  angle  S''SL  will  become 
larger,  and  a  second  bearing  must 
be  taken  at  the  moment  when 
the  ship  reaches  the  point  S', 
where  the  angle  S"SL  has  become 
S'^S'L.  This  point  S'  must  be 
so  chosen  that  the  angle  S'^S'L 
is  just  twice  the  angle  S'^SL  ob- 
served at  S ;  or,  in  this  case,  44°. 
This  is  called  ''doubling  the  bear- 
FiG.  9.  —  Ship's  Position  by  Two  ing  from  the  bow,"  and  it  can 
eanngs.  easily  be  accomplished  if  we  con- 

tinue watching  the  compass  bearing  of  L  as  the  ship  goes 
ahead,  and  catch  the  observation  at  the  right  moment.  The 
ship's  course  not  having  been  changed  from  20°  (this  is 
important),  the  right  moment  will  occur  when  L  bears 
20°  +  44°  =  64°  by  the  compass. 

It  can  easily  be  proved  by  geometry  that  the  distance 
S^L  between  the  ship  at  S'  and  the  lighthouse  at  L  will  be 
equal  to  the  distance  SS'  traveled  by  the  ship  in  the  inter- 
val between  the  two  observations.  This  distance  can  be 
estimated  quite  accurately  with  an  instrument  called  a 
''log,"  or  "patent  log,"  which  is  towed  astern  of  the  ship. 
It  is  so  constructed  that  it  turns  as  it  is  pulled  through  the 
water,  and  the  number  of  turns  is  automatically  counted  by 
an  attached  contrivance  on  deck.  The  count  is  (also  auto- 
matically) turned  into  miles  of  distance ;  so  that  the  log  on 
deck  will  indicate  how  far  the  ship  traveled  from  *S  to  *S'. 


COASTWISE  NAVIGATION  55 

As  soon  as  we  know  the  distance  S'L  and  the  bearing  of 
the  Hne  S'L,  we  can  ''lay  down"  or  ''plot"  the  position  of 
*S'  on  the  chart ;  and  this  will  be  a  "good  fix."  To  do  this, 
let  us  indicate  by  B'  the  bearing  of  the  line  S'L,  and  then 
draw  on  the  chart,  through  the  lighthouse  L,  a  pencil  line 
whose  bearing  from  L  is  B'  -\-  180°,  or  ^'B'  reversed."  This 
can  be  done  with  a  "course  protractor,"  or  with  "parallel 
rulers,"  instruments  to  be  purchased  from  any  dealer  in 
navigators'  supplies.  Next  we  measure  or  "lay  off"  on  that 
line  the  distance  S'L,  equal  to  the  run  SS'  as  it  came  from 
the  log.  We  always  know  the  right  "scale"  of  the  chart 
(or  fraction  of  an  inch  corresponding  to  one  logged  mile) 
which  must  be  used  in  laying  off  the  distance  S'L;  for  we 
know  that  one  mile  always  corresponds  to  1  minute  of 
latitude  (p.  15),  and  the  right-  and  left-hand  edges  of  the 
chart  are  always  divided  into  degrees  and  minutes  of  latitude. 

Since  the  above  bearings  were  observed  by  compass,  it 
is  now  important  to  consider  the  compass  error  (p.  43). 
This  will  not  affect  the  observations,  because  it  will  be  the 
same  for  both  ship's  course  and  lighthouse  bearing,  so  the 
angles  S"SL  and  S"S'L,  which  are  obtained  by  subtraction, 
will  be  the  same  as  if  there  were  no  compass  error.  But 
when  we  come  to  plotting  on  the  chart,  the  compass  bearing 
B'  must  be  corrected  by  adding  the  deviation  from  the 
deviation  table  (pp.  48,  49).  The  resulting  magnetic  bear- 
ing (p.  49)  must  be  used  for  B',  if  the  chart  has  printed 
on  it  a  compass  card  (p.  41)  showing  magnetic  bearings. 
If  the  printed  card  shows  true  bearings  only,  B'  must  be 
corrected  for  both  deviation  and  variation  (p.  43). 

A  specially  important  case  of  the  foregoing  occurs  when 
the  two  angles  S"SL  and  S"S'L  are  45°  and  90°.  The 
second  bearing  B'  will  then  put  the  light  just  abeam,  and 
the  distance  by  log,  *S*S',  is  the  distance  at  which  the  ship 
passes  the  light  abeam.  This  case  is  called  a  "bow-and- 
beam  bearing."  The  navigator  sights  the  light  when  it  bears 
45°  or  4  points  (p.  52)  "broad"  on  the  bow,  "starboard," 


56  NAVIGATION 

or  ''port."  He  then  ''reads"  the  log.  When  he  brings 
the  Hght  abeam  through  the  motion  of  the  ship,  he  reads 
the  log  again,  and  the  run  in  the  interval,  as  taken  from  the 
log,  is  the  light's  distance  abeam. 

When  sailing  along  the  coast,  it  is  particularly  important 
so  to  shape  the  ship's  course  that  lights  and  other  promi- 
nent landmarks  will  be  passed  at  the  right  distance  abeam. 
The  chart  shows  what  the  right  distance  is :  if  the  navigator 
shapes  a  course  which  makes  the  distance  abeam  too  small, 
he  may  fail  to  clear  rocks  or  shoals  extending  seaward ;  and 
if  he  makes  it  too  large,  he  may  lengthen  his  voyage  unneces- 
sarily in  rounding  the  light. 

There  are  certain  pairs  of  angles  {S"SL  and  S"S'L)  which 
will  make  known  the  coming  distance  abeam  long  before 
the  ship  is  dangerously  near  the  light.  These  angles,  S"SL 
and  S"S'L,  are  called  "bearings  from  the  bow"  (see  p.  54), 
since  they  are  really  measured  from  the  ship's  bow  instead 
of  the  north.  If  the  two  bearings  from  the  bow  are  either 
of  the  following  pairs  : 

22°  and  34°,  32°  and  59°, 

27°  and  46°,  40°  and  79°, 

then  the  logged  distance  in  the  interval  between  the  two 
observations  is  the  distance  at  which  the  ship  will  pass  the 
light  abeam  if  she  continues  on  her  present  course.  This 
kind  of  observation  will  inform  the  navigator  whether  his 
course  is  safe  in  ample  time  to  change  it  if  necessary ;  and, 
since  in  this  case  no  bearings  are  marked  on  the  chart,  no 
attention  need  be  paid  to  compass  error. 

When  two  or  more  known  and  conspicuous  landmarks 
are  visible  from  the  ship,  it  is  possible  to  secure  a  fix  by 
means  of  "cross-bearings."  Observe  the  bearings  of  the 
objects  as  nearly  simultaneously  as  possible.  Allow  for 
compass  error  in  the  manner  just  explained.  Calculate 
for  each  object  a  reversed  bearing  by  adding  180°  to  its 
observed  bearing.     Draw  on  the  chart  through  each  object 


COASTWISE  NAVIGATION 


57 


a  pencil  line  having  the  proper  reversed  bearing  and  these 
lines  will  intersect  at  the  point  on  the  chart  where  the  ship 
is  located.  Figure  10 
illustrates  this  matter. 
L,  U,  L"  are  Hghts  or 
landmarks  ashore, 
visible  from  the  ship, 
and  also  printed  on 
the  chart.  The  ship 
is  at  S.  The  lines  in- 
tersecting at  S  repre- 
sent  the  reversed 
bearings  of  L,  U ,  L" , 
as  observed  from  S. 
Only  two  lines  are  nec- 
essary ;  and  they 
should  be  chosen  so 
that  the  angle  be- 
tween them  is  as  near 


Fig,  10.  —  Ship's  Position  by  Cross  Bearings. 


a  right  angle  as  possible,  if  high  accuracy  is  required  in  the 
fix.  The  third  object  and  line  merely  serve  as  an  additional 
check  or  safeguard  against  error. 

In  addition  to  the  foregoing  methods  of  locating  a  ship 
by  observations  of  objects  ashore,  there  is  a  way  to  avoid 
sunken  rocks  or  shoals  without  actually  locating  the  ship 
on  the  chart.  It  is  called  the  ''danger  angle,"  and  is  shown 
in  Fig.  11.  The  small  circle  is  supposed  drawn  on  the  chart 
around  a  rocky  shoal  K  which  must  be  cleared  by  the  ship 
traveling  along  the  course  SS' .  To  make  certain  of  clearing 
it  safely,  the  navigator  selects  two  visible  objects  ashore, 
and  shown  on  the  chart  at  L  and  U .  He  draws  on  the 
chart  a  large  circle  passing  through  L  and  L',  and  just  touch- 
ing the  dangerous  small  circle  at  T.  There  is  no  difficulty 
in  finding  the  center  of  the  large  circle,  because  it  must  be 
somewhere  on  the  line  PQ,  which  is  drawn  at  right  angles 
to  the  line  LU  at  its  middle  point  P.     A  few  trials  with  a 


58 


NAVIGATION 


pair  of  compasses  will  locate  the  center.  Next,  the  two  lines 
LT  and  L'T  are  drawn.  Then  the  angle  LTU  is  called  the 
danger   angle. 

Now  it  is  a  principle  of  geometry  that  if  we  select  other 
points  on  the  large  circle,  such  as  T'  and  T",  the  angles 


Fig.  11.  —  The  Danger  Angle. 

LT'U ,  LT"U,  etc.,  will  all  be  equal,  and  will  contain  the 
same  number  of  degrees  as  the  danger  angle  LTU.  It  fol- 
lows that  if  the  navigator  measures  from  the  deck  the  angle 
formed  by  two  lines  drawn  to  the  ship  from  L  and  U,  and 
if  he  finds  it  equal  to  the  danger  angle  LTU,  as  measured 
on  the  chart  with  a  protractor  (p.  55),  he  then  knows  that 
the  ship  is  somewhere  on  the  large  circle,  and  is  therefore 
perhaps  too  near  the  small  dangerous  circle.  If,  on  the 
other  hand,  the  ship  is  entirely  outside  the  large  circle,  and 
therefore  surely  safe  from  the  dangers  of  the  small  circle, 


COASTWISE   NAVIGATION 


59 


the  measured  angle  at  the  ship  between  the  objects  L  and 
L'  will  always  be  smaller  than  the  danger  angle  LTU. 

Angles  can  be  measured  from  the  deck  by  taking  compass 
bearings  of  L  and  U.  The  difference  of  the  two  will  be  the 
deck  angle,  which  should  be  smaller  than  the  danger  angle 
measured  on  the  chart.  But  the  very  best  way  to  measure 
the  deck  angle  is  to  use  the  sextant,  an  angle-measuring 
instrument  to  be  described  later  (p.  61). 

The  danger  angle  can  also  be  used  when  it  is  necessary  to 
pass  between  a  sunken  danger  circle  and  the  shore.  The 
large  circle  is  then  drawn  through  L  and  U  as  before,  but  in 
such  a  way  as  just  to  touch  the  inside  of  the  small  circle 
instead  of  the  outside.  To  pass  inshore  of  the  small  circle 
it  is  then  necessary  for  the  navigator 
to  keep  his  measured  deck  angle  larger 
than  the  danger  angle,  instead  of 
smaller. 

Navigators  also  use  at  times  a 
means  of  safety  known  as  the  ''danger 
bearing,"  illustrated  in  Fig.  12. 
There  is  but  one  charted  object  in 
sight  ashore  at  the  point  L.  The  ship 
at  S  must  steer  in  such  a  way  as  to 
avoid  sunken  rocks  at  K.  Evidently, 
she  must  pass  outside  the  line  SQ,  of 
which  the  bearing  from  the  north  is 
the  angle  NSQ,  which  can  be  meas- 
ured on  the  chart.  This  is  the  danger 
bearing,  and  the  ship's  course  aS*S',  to 

be  safe,  must  be  greater  than  the  danger  bearing.  In  the 
case  shown  in  the  figure,  the  danger  bearing  would  be  very 
useful  long  before  a  fix  could  be  had  by  means  of  bearings 
from  the  bow  or  bow-and-beam  bearings. 

Finally,  to  complete  this  part  of  our  subject,  it  is  neces- 
sary to  mention  ''soundings,"  which  are  a  method  of  feeU 
ing  the  land,  even  when  it  cannot  be  seen.     By  means  of 


Fig.  12.  — The  Danger 
Bearing. 


60  NAVIGATION 

the  *' lead-line"  the  mariner  can  ascertain  when  he  is  in 
shoal  water ;  and  as  depths  of  water  are  always  marked  on 
the  chart,  he  can  often  get  valuable  information  as  to  the 
ship's  position.  As  she  runs  along  her  course,  he  can  take 
a  ''line  of  soundings"  and  upon  examining  the  chart  he 
will  often  find  but  a  single  possible  line  on  the  chart  where 
the  charted  depths  correspond  with  those  observed.  It 
follows  that  the  ship's  course  must  have  been  along  that 
line  on  the  chart ;  and  at  an  anxious  moment,  in  a  fog,  such 
a  check  will  be  a  great  relief  to  the  navigator.  Even  in 
the  ocean,  far  from  land,  it  is  possible  to  take  soundings 
with  the  ''sounding  machine"  at  great  depths,  and  in  some 
parts  of  the  ocean  quite  accurate  locating  of  the  ship  will 
result.  Specimens  from  the  ocean  floor  can  also  be  brought 
up  by  attaching  some  sticky  grease  to  the  bottom  of  the 
lead,  and  at  times  these  specimens  also  give  information 
of  value,  for  the  charts  always  specify  the  kind  of  bottom 
existing  in  various  parts  of  the  ocean. 


CHAPTER   VI 
THE  SEXTANT 

We  have  twice  made  reference  to  this  instrument  —  once 
(p.  5)  as  a  contrivance  for  ascertaining  by  observation  how 
high  the  sun  is  in  the  sky,  and  again  (p.  59)  in  the  measure- 
ment of  the  danger  angle.  These  two  uses  of  the  sextant 
are  not  inconsistent,  for  it  is  really  intended  for  the  measure- 
ment of  any  angle  (p.  8)  formed  at  the  observer's  eye  by 
two  lines  drawn  to  two  distant  objects.  In  the  case  of  the 
danger  angle  these  two  distant  objects  are  landmarks 
ashore;  in  the  case  of  the  sun  they  are  the  ''horizon"  line 
(where  sea  and  sky  seem  to  meet),  and  the  sun  itself.  This 
height  of  the  sun  (or  of  any  star)  in  the  sky  is  called  its 
''altitude";  and  so  the  altitude  is  always  an  angle,  to  be 
measured  in  degrees  and  minutes.  The  point  directly  over- 
head is  the  "zenith";  the  angle  between  lines  drawn  to 
horizon  and  zenith  is  90°,  or  a  right  angle.  An  altitude  of 
40°,  for  instance,  simply  means  that  the  distance  from  the 
horizon  to  the  sun  is  f^  of  the  total  distance  from  horizon 
to  zenith. 

Figure  13  will  give  an  idea  of  the  construction  of  the  sex- 
tant.^ The  essential  parts  are  two  small  silvered  mirrors, 
M  and  m;  a  telescope,  EK;  and  a  circle,  A  A,  engraved 
with  "graduations,"  by  means  of  which  angles  may  be 
measured  upon  it  in  degrees,  minutes,  and  seconds.  The 
mirror  m  and  the  telescope  EK  are  firmly  attached  to  the 
sextant ;  but  the  mirror  M  is  pivoted  in  such  a  way  that  it 

1  Quoted  in  part  from  Jacoby's  "Astronomy,  a  Popular  Hand- 
book," Macmillan,  1913 ;  reprinted  1915. 

61 


62 


NAVIGATION 


can  be  turned,  and  the  angle  through  which  it  is  turned 
measured  on  the  circle  by  means  of  the  index  CB.  When 
the  mirror  M  is  turned  until  it  is  parallel  to  the  fixed  mirror 
m,  the  circle  ''reads"  or  indicates  0°,  because  the  angle  be- 
tween the  two  mirrors  is  then  0°.     In  all  other  positions 


Fig.  13.  — The  Sextant. 


of  the  mirror  M  the  circle  measures  the  angle  between  the 
two  mirrors.  P  and  Q  are  sets  of  colored  glasses,  which  can 
be  interposed  temporarily,  when  the  sun's  rays  are  so  bril- 
liant as  to  be  hurtful  to  the  observer's  eye.  R  is  a  small 
magnifying  glass,  pivoted  at  S,  intended  to  facilitate  the 
examination  of  the  index  CB.  At  C  and  B  are  shown  the 
''clamp,"  by  which  the  index  can  be  fastened  to  the  circle, 
and  the  "tangent  screw,"  or  "slow-motion  screw"  which 
will  adjust  it  delicately,  after  it  has  been  clamped.  I  and  F 
are  additional  telescopes  or  accessories. 

The  mirror  m  has  an  important  peculiarity.  The  silver- 
ing is  scraped  away  at  the  back  of  the  mirror  from  half  its 


THE   SEXTANT  63 

surface.  Thus  only  one  half  reflects ;  the  other  half  is 
simply  transparent  glass.  A  navigator  looking  into  the 
telescope  at  E  will  therefore  look  through  the  mirror  m  with 
half  his  telescope,  and  with  the  other  half  he  will  look  into 
the  mirror. 

Now  it  is  a  fact  that  half  a  telescope  acts  just  like  a  whole 
one.  If  a  person  using  an  ordinary  spy-glass  half  covers 
the  big  end  with  his  hand,  he  will  see  the  same  view  he  saw 
with  the  whole  glass.  Only,  as  half  the  ''light-gathering" 
power  is  cut  off,  this  view  will  be  fainter,  —  less  luminous. 
Applying  this  to  the  sextant  telescope,  it  is  clear  that  the 
observer  will  see  two  things  at  once  :  with  half  the  telescope 
he  will  see  what  is  visible  through  the  mirror  m ;  and  with 
the  other  half  he  will  see  what  is  visible  by  reflection  from 
the  mirror  m. 

If  he  holds  the  sextant  in  such  a  position  that  the  telescope 
is  horizontal,  while  the  frame  of  the  instrument  is  vertical, 
he  will  see  the  visible  sea  horizon  with  half  the  telescope 
through  the  mirror  m.  If  the  other  mirror  M  is  then  turned 
to  the  proper  position,  it  is  possible  to  see  the  sun  in  the  sky 
at  the  same  time,  with  the  other  half  of  the  telescope,  the 
solar  rays  having  been  reflected  successively  from  both  mir- 
rors, M  and  m.  To  make  this  possible,  the  sextant  tele- 
scope must  be  aimed  at  that  point  of  the  sea  horizon  which 
is  directly  under  the  sun.  The  solar  rays  will  then  strike 
the  mirror  M  first ;  be  thence  reflected  to  the  silvered  part 
of  the  mirror  m ;  and  finally  reflected  a  second  time  into 
the  telescope.  Therefore  the  observation  consists  in  so 
turning  the  movable  mirror  M,  that  the  sun  and  horizon 
can  be  seen  coincidently  in  the  telescope. 

The  angle  between  the  mirrors  can  then  be  measured 
on  the  circle ;  and  it  is  easy  to  prove  by  geometry  that  the 
angular  altitude  of  the  sun  will  be  twice  the  angle  between 
the  two  mirrors.  Thus  it  should  merely  be  necessary  to 
double  the  mirror  angle,  as  indicated  by  the  sextant  index, 
to  obtain  the  solar  altitude.     But  the  sextant  makers  always 


64  NAVIGATION 

save  the  navigator  the  trouble  of  doubling  the  angle  by  the 
simple  device  of  numbering  half  degrees  on  the  arc  AA  as 
if  they  were  whole  degrees ;  so  the  angle  as  it  comes  from  the 
sextant  is  already  doubled  for  further  use.  The  mirror  m 
is  called  the  ''horizon  glass,"  because  the  navigator  looks 
through  it  at  the  horizon.  The  other  mirror  M  is  the  ''index 
glass,"  because  it  is  attached  to  the  index  arm. 

When  the  sextant  is  used  for  non-astronomical  observa- 
tions, such  as  the  danger  angle,  the  frame  is  held  horizontally, 
instead  of  vertically,  as  in  observations  of  the  sun.  The 
telescope  is  aimed  at  the  left-hand  object  ashore,  and  that 
object  is  viewed  through  the  horizon  glass  m.  The  index 
glass  M  is  then  turned  until  light  from  the  right-hand  object 
is  also  brought  into  the  telescope,  after  successive  reflections 
from  the  two  mirrors  M  and  m.  The  two  objects  will  then 
be  seen  "superposed,"  and  the  sextant  arc  will  give  the 
angle  between  two  lines  drawn  from  the  observer  on  board 
to  the  two  objects  ashore.  This  angle  should  be  smaller 
than  the  danger  angle  to  keep  the  ship  safely  off-shore  of 
sunken  dangers  (p.  59). 

Reading  the  sextant  circle,  or  ascertaining  from  it  the 
angle  that  has  been  measured,  is  accomplished  by  means  of 
a  "vernier."  This  is  a  short  circular  arc,  engraved  with 
graduations  resembling  those  on  the  sextant  circle,  attached 
to  the  index  CB  (fig.  13)  just  under  the  little  magnifier  R. 
It  is  so  placed  that  the  graduations  on  the  sextant  circle 
and  the  vernier  are  close  together  and  can  be  seen  at  the 
same  time  through  the  magnifier  R.  Figure  14  gives  an  idea 
of  the  vernier  and  a  part  of  the  sextant  circle  near  the  zero 
of  its  graduations.  Numbers  on  both  circle  and  vernier 
increase  toward  the  left.  On  the  circle,  the  largest  spaces, 
marked  by  long  lines,  are  whole  degree  spaces.  Each  is 
usually  divided  into  two  halves  of  30'  each  indicated  by 
shorter  lines,  and  these  are  again  subdivided  into  three 
small  spaces  of  10'  each.  The  divisions  on  the  vernier 
resemble  those  on  the  circle,  except  that  the  degree  spaces 


THE   SEXTANT 


65 


of  the  former  are  here  called  min- 
ute spaces,  and  the  10'  spaces  of 
the  former  are  called  10"  spaces. 
The  real  index  of  the  instru- 
ment is  the  zero  mark  on  the 
vernier,  sometimes  provided  with 
an  engraved  ''arrow."  If  this 
falls  exactly  on  a  degree  mark  of 
the  circle,  say  the  1°  mark,  the 
reading  of  the  instrument  is  ex- 
actly 1°  0'  0".  If  it  falls  exactly 
on  a  small  line  of  the  circle,  say 
the  second  to  the  left  of  the  1° 
mark,  the  reading  is  exactly  1° 
20'  0".  But  if  it  falls  between  two 
of  the  small  lines,  say  between 
the  20'  and  30'  marks  to  the  left 
of  the  1°  mark  (as  shown  in  the 
figure),  the  reading  must  be  1° 
20'  and  a  "hiU'  It  is  the  busi- 
ness of  the  vernier  to  estimate 
the  size  of  that  bit.  To  do  this 
look  along  the  vernier  until  you 
find  a  line  which  is  exactly  op- 
posite some  line  on  the  circle. 
There  will  always  be  such  a  line  : 
in  the  figure  it  is  the  6'  line  of  the 
vernier.  Pay  no  further  atten- 
tion to  noting  which  line  on  the 
circle  is  the  one  thus  "exactly 
opposite";  it  matters  not  which 
line  it  is.  But  read  carefully  the 
number  on  the  vernier  belonging 
to  the  ''exactly  opposite"  line 
you  have  found  there.  Being  on 
this  occasion  the  6'  line,  it  follows 


c 
to 


lO 


a> 


> 


1— t 

6 


66  NAVIGATION 

that  the  bit  is  6' ;  and  as  we  found  the  reading  to  be  1°  20' 
and  a  bit,  the  complete  reading  is  1°  20'  +  6'  =  1°  26'. 

If  the  vernier  line  that  happened  to  be  ''exactly  opposite" 
was  not  one  of  the  ten  long  minute  lines,  but  fell  between 
two  of  them,  it  would  indicate  that  the  bit  was  made  up  of 
minutes  and  seconds,  instead  of  being  an  exact  number  of 
minutes.  For  each  space  the  ''exactly  opposite"  vernier 
line  happens  to  lie  to  the  left  of  a  long  vernier  minute  line, 
10''  must  be  added  to  the  bit.  For  instance,  if  in  the  figure 
the  "exactly  opposite"  vernier  line  was  the  next  short  one 
to  the  left  of  the  6'  long  line,  the  bit  would  be  6'  10",  and  the 
complete  reading  1°  26'  10",  instead  of  1°  26'.  But  seconds 
are  not  really  required  when  observing  aboard  ship,  so  that 
it  will  be  sufficient,  in  using  the  vernier,  to  find  the  number 
of  the  long  vernier  line  that  comes  nearest  to  being  "exactly 
opposite." 

It  will  also  be  noticed  in  the  figure  that  the  sextant  circle 
has  some  additional  graduations  to  the  right  of  the  0°  mark. 
These  are  called  "off  the  arc"  graduations,  and  it  is  some- 
times necessary  to  read  a  small  angle  upon  them,  measuring 
from  the  0°  mark  to  the  right  instead  of  the  left.  This  makes 
it  necessary  to  read  the  vernier  backwards,  calling  the  0' 
mark  of  the  vernier  10'  and  the  10'  mark  0'. 

This  backward  reading  of  the  vernier  offers  no  particular 
difficulty,  and  it  is  especially  useful  in  determining  by  ob- 
servation the  "index  error"  of  the  sextant.  We  have  seen 
(p.  62)  that  when  the  two  sextant  mirrors  are  parallel, 
the  index  should  read  0°  0'  0".  But  it  is  seldom  possible 
to  adjust  the  instrument  so  that  this  condition  will  be  satis- 
fied exactly ;  nor  would  the  adjustment  remain  perfect  very 
long.  A  better  plan  is  to  determine  by  observation  how 
much  the  reading  differs  from  0°  0'  0",  when  the  mirrors 
are  parallel.  This  difference  is  the  index  error,  and  must 
be  applied  as  a  correction  to  all  angles  observed  with  the 
instrument. 

It  is  easy  to  make  the  mirrors  parallel :   we  have  merely 


THE   SEXTANT  67 

to  sight  some  distant  well-defined  terrestrial  object  like  the 
gilt  ball  on  the  top  of  a  flagpole  (or  the  sea  horizon,  if  aboard 
ship  at  sea),  after  clamping  the  index  near  0°.  We  shall 
then  see  in  the  telescope  two  images  of  the  distant  object; 
one  by  direct  vision  through  the  unsilvered  part  of  the  hori- 
zon glass,  the  other  after  reflection  from  both  mirrors.  By 
means  of  the  tangent  screw,  the  observer,  with  his  eye  at 
the  telescope,  can  bring  these  two  images  together,  so  that 
they  will  appear  as  a  single  image.  Then  the  mirrors  will 
be  parallel,  and  the  vernier  should  read  0°  0'  0".  If  it  actually 
reads  0°  8',  for  instance,  instead  of  0°  0'  0",  it  means  that  the 
reading  is  8'  too  large  on  account  of  index  error ;  and  every 
angle  measured  with  that  sextant  at  that  time  will  be  8' 
too  large,  and  must  be  corrected  by  subtracting  8'  from  it. 

If,  on  the  other  hand,  the  reading  is  8'  ''off  the  arc," 
when  it  should  be  0°  0',  the  instrument  reads  8'  too  small, 
and  any  angle  measured  with  it  must  be  corrected  by  adding 
8'  to  it. 

For  accurate  determination  of  the  index  error  (and  it 
should  be  checked  frequently),  navigators  prefer  to  observe 
the  sun,  or  at  night,  a  star.  If  a  star  is  used,  the  process 
is  the  same  as  just  described  for  a  flagpole  ball.  But  if 
the  sun  is  used,  a  slightly  different  method  is  required.  The 
sun,  as  seen  in  the  telescope,  shows  a  round  disk  of  con- 
siderable size,  and  it  is  not  possible  to 
superpose  the  two  images  accurately. 
Therefore  it  is  better  to  make  them 
just  touch,  as  shown  in  Fig.  15,  when 
they  are  said  to  be  ''tangent"  to  each 
other.  This  must  be  done  successively 
in  two  positions,  AB  and  BA.  In 
other  words,  after  the  first  " tangency "  ^^'  •—  ^  ^^  ""or. 
has  been  observed,  the  tangent  screw  (B,  fig.  13)  is  manipu- 
lated until  the  image  A  passes  across  B  from  top  to  bottom, 
and  gives  a  new  tangency  in  the  second  position. 

Each  tangency  will  give  a  reading  of  the  vernier.     Unless 


68  NAVIGATION 

the  sextant  is  greatly  out  of  adjustment,  one  of  these  read- 
ings will  be  off  the  arc,  the  other  on  the  arc.  If  there  were 
no  index  error,  the  off-arc  and  on-arc  readings  would  be 
equal;  if  they  differ,  half  the  difference  is  the  index  error. 
If  the  off-arc  reading  is  the  larger,  all  altitudes  measured 
with  that  sextant  must  be  increased  by  the  amount  of  the 
index  error ;  and  if  the  on-arc  reading  is  the  larger,  all  such 
altitudes  must  be  similarly  diminished. 

The  following  is  an  example  of  an  index  error  determina- 
tion : 

On-arc  readings,  Off-arc  readings, 
31'  20"  33'  20" 

31  40  33  50 

30  50  34     0 

Means,    31'  17"  33'  43" 

The  difference  is  33'  43"  -  31'  17"  =  2'  26".  Half  the 
difference,  or  1'  13",  is  the  index  error ;  and  because  readings 
on  the  arc  are  the  smaller,  all  angles  read  with  this  instru- 
ment must  be  increased  by  1'  13",  or,  for  ordinary  purposes 
of  navigation,  by  1'. 

In  addition  to  certain  ''adjusting  screws"  with  which 
the  index  error  can  be  reduced  when  it  becomes  unduly 
large,  means  are  provided  for  three  other  sextant  adjust- 
ments.    These  are  : 

1.  To  make  the  index  glass  perpendicular  to  the  frame  of 
the  instrument. 

2.  To  do  the  same  with  the  horizon  glass. 

3.  To  set  the  telescope  parallel  to  the  frame  of  the  instru- 
ment. 

These  adjustments  are  always  completed  by  the  maker 
before  a  sextant  is  sent  out,  nor  does  the  navigator  usually 
need  to  correct  them  himself.  But  it  is  important  to  know 
how  to  test  them  occasionally.  Perpendicularity  of  the 
index  glass  can  be  examined  by  looking  into  the  glass  very 
obliquely  with  the  index  set  near  0°.  It  is  then  possible  to 
see  the  inner  edge  of  the  sextant  circle  both  by  looking  at 


THE   SEXTANT  69 

it  directly,  past  the  edge  of  the  index  glass,  and  also  by  reflec- 
tion in  the  glass  itself.  The  inner  edge  of  the  circle  should 
form  a  continuous  line  when  so  examined,  if  the  glass  is 
perpendicular ;  but  if  it  is  inclined,  the  line  will  appear  broken, 
instead  of  continuous. 

Secondly,  perpendicularity  of  the  horizon  glass  can  be 
tested  at  the  same  time  the  index  error  is  determined  by 
observing  a  star  or  a  distant  terrestrial  point  (p.  67).  The 
index  glass  having  been  properly  adjusted  to  perpendic- 
ularity, the  two  mirrors  can  never  be  made  parallel  by 
moving  the  index,  unless  the  horizon  glass  is  also  properly 
perpendicular.  Any  existing  lack  of  adjustment  will  there- 
fore betray  itself  in  the  index  error  determination,  because 
the  two  images  of  the  star  or  distant  object  will  not  be  super- 
posed in  any  position  of  the  index. 

Thirdly,  the  parallelism  of  the  telescope  to  the  frame  of 
the  instrument  can  usually  be  best  tested  with  an  ordinary 
pair  of  '^calipers." 

Having  thus  described  the  sextant,  its  adjustments,  and  its 
use  from  the  deck,  we  have  still  to  explain  how  it  can  be  used 
ashore.    Sometimes  it  is  necessary  for  the    s, 
navigator  to  make  observations  ashore, 
when  it  is  not  usually  possible  to  see  the 
horizon  line    (p.   61).      Recourse    must 
then  be  had  to  an  ''artificial  horizon, '^ 
which   is   simply  an   iron  basin  full  of 
mercury  covered  with  a  glass  roof.     The 
mercury   furnishes  an   almost  perfectly 
horizontal   mirror,    and   the   glass    roof 
prevents  wind  from  ruflfling  the  mercury 
surface,  and  thus  destroying  the  mirror,     fiq  le.  —  Artificial' 
Figure  16  explains  the  principle  of  the  Horizon, 

artificial  horizon.  HH  is  the  mercury  mirror,  S  the  sun, 
and  X  the  sextant.  The  observer  aims  the  sextant  telescope 
at  the  mercury  where  he  can  see  a  reflection  of  the  sun.  He 
then  measures  with  the  instrument  the  angle  between  a  line 


70 


NAVIGATION 


drawn  to  the  sun  as  seen  reflected  in  the  mercury  and  another 
line  drawn  to  the  actual  sun  in  the  sky.  It  can  be  shown 
by  geometry  that  this  measured  angle  will  be  just  twice  the 
real  altitude  of  the  sun,  such  as  it  would  be  if  observed  from 
the  sea  horizon.  Therefore,  in  using  the  artificial  horizon, 
it  is  merely  necessary  to  divide  the  sextant  angle  by  2  to  ob- 
tain the  correct  altitude  of  the  sun. 

In  observations  of  this  kind  two  ''suns"  are  seen  at  the 
same  time  in  the  telescope,  just  as  is  the  case  in  index  error 
observations  (p.  67) ;  whereas  in  observing  from  the  sea 
horizon,  the  telescope  shows  only  one  solar  image  and  the 
horizon  line.  When  there  are  thus  two  solar  images,  they 
must  be  brought  into  tangency,  just  as  we  have  already 
explained  for  index  error  (p.  67).  When  there  is  but  one, 
it  must  be  brought  into  tangency  with  the  visible  sea 
horizon  line. 

But  this  altitude  is  not  yet  ready  to  be  used  in  the  further 
calculations  for  obtaining  the  position  of  the  ship  in  latitude 

and  longitude.  Further  pre- 
paratory corrections  must  be 
applied,  in  addition  to  the 
index  error  (p.  66),  which  is 
always  the  first  correction  to 
receive  attention .  These  pre- 
paratory corrections  are : 

1.  ''Dip"  of  the  sea  hori- 
zon, due  to  the  elevation  of 
the  navigator  on  the  ship's 
deck  above  the  surface  of  the 
sea.     Its  cause  is  shown  in 
Fig.  17.   C  is  the  center  of  the 
earth,  K  a  point  at  sea  level, 
and  0  the  navigator,  elevated 
a  distance  OK  above  the  sea.    OZ  is  the  direction  of  the  ze- 
nith (p.  61),  OS  the  direction  of  the  sun,  and  OH  a  horizontal 
line  from  0.     OT  is  a  line  drawn  through  0,  and  just  touch- 


FiG.  17.  —  Dip  of  the  Horizon. 


THE   SEXTANT  71 

ing  the  sea  surface  at  T'.  Evidently  OT  will  be  the  direc- 
tion of  the  sea  horizon,  where  sky  and  sea  seem  to  meet. 
Therefore,  the  altitude  of  the  sun,  as  measured  from  the 
visible  sea  horizon,  will  be  the  angle  SOT ;  whereas  the  angle 
we  require  is  the  angle  SOH,  or  the  altitude  of  the  sun 
above  the  true  horizontal  line  OH.  Therefore  the  angle 
HOT  is  a  correction  for  dip  which  must  be  subtracted  from 
all  measured  altitudes,  and  the  amount  of  the  correction 
depends  on  the  height  of  the  navigator's  eye  above  the  sea 
surface. 

2.  *' Refraction"  is  a  bending  of  the  light  rays  as  they 
come  down  to  us  from  the  sun  through  the  terrestrial  atmos- 
phere. It  always  makes  the  sun  seem  higher  in  the  sky 
than  it  really  is,  giving  another  subtractive  correction  for 
the  observed  altitude.  The  bending  here  involved  is  due 
to  the  passage  of  the  sun's  light  rays  through  atmospheric 
strata  of  increasing  density  as  the  light  approaches  the 
earth's  surface. 

3.  '^ Parallax"  is  a  small  correction  which  must  be  added 
to  the  observed  altitude  of  the  sun.  In  strict  theory,  all  astro- 
nomic observations  are  supposed  to  be  made  from' the  earth's 
center  instead  of  its  surface  where  the  ship  floats ;  and  the 
small  parallax  correction  allows  for  this  minor  theoretic 
point.  In  the  case  of  star  observations  this  correction  is 
zero. 

4.  ''Semidiameter"  is  a  correction  depending  on  the 
choice  by  the  navigator  of  a  particular  point  on  the  sun's 
disk  (p.  67)  for  observation.  The  sun's  altitude,  as  used 
in  the  further  calculations,  should  be  the  altitude  of  the  sun's 
center ;  but  it  is  impossible  to  locate  the  center  of  the  disk 
accurately  in  the  telescope,  so  the  navigator  always  observes 
the  lowest  point  of  the  disk.  This  is  called  the  '^  lower 
limb"  of  the  sun. 

Beginners  sometimes  have  difficulty  in  distinguishing 
the  upper  from  the  lower  limb  in  the  telescope.  The  best 
way  to  do  this  is  to  focus  the  telescope  on  some  distant 


72  NAVIGATION 

object,  and  note  whether  it  appears  upside-down  in  the 
field  of  view.  If  so,  the  telescope  is  an  '' inverting"  one, 
and  the  top  of  the  sun  must  be  observed,  as  it  appears  in 
the  telescope,  though  it  will  really  be  the  correct  (or  lower) 
limb,  because  of  inversion  by  the  telescope.  When  using  the 
artificial  horizon  with  an  inverting  telescope,  the  tangency 
must  be  made  by  bringing  the  bottom  of  the  mercury  image 
in  contact  with  the  top  of  the  other  image.  The  high-pow> 
ered  telescopes  supplied  with  good  sextants  are  usually  in- 
verting telescopes. 

Evidently  the  measured  altitude,  as  it  comes  from  the 
sextant,  must  be  increased  by  the  amount  by  which  the  sun's 
center  is  higher  than  the  lower  limb,  and  this  is  the  sun's 
semidiameter.  The  index  correction,  together  with  the 
above  four  additional  corrections,  will  fully  prepare  a  meas- 
ured sextant  altitude  of  the  sun  for  further  use  in  naviga- 
tional calculations.  In  the  case  of  a  star,  which  appears 
in  the  telescope  as  a  point  of  light  only,  without  any  per- 
ceptible disk,  no  semidiameter  or  parallax  corrections  are 
required;  and  in  using  the  artificial  horizon  (p.  69),  no 
correction  for  dip  is  necessary,  either  for  the  sun  or  a  star. 

It  is  possible  to  arrange  these  various  corrections  in  con- 
venient tables.  Thus,  in  Table  6  (p.  247),  we  give  a  combi- 
nation of  corrections  2  (refraction),  3  (parallax),  and  4  (semi- 
diameter), to  be  used  for  observations  of  the  sun's  lower 
limb,  and  the  same  combination  without  the  semidiameter 
and  parallax  ^  to  be  used  for  star  observations.  It  will  be 
noticed  that  the  tabular  corrections  vary  for  different  values 
of  the  observed  altitude,  which  appears  in  the  left-hand  col- 
umn of  the  table.  This  variation  comes  mainly  from  the 
refraction  part  of  the  combined  correction,  for  the  refrac- 
tion is  much  greater  when  the  sun  or  star  is  observed  at  a 
low^  altitude  near  the  horizon  than  it  is  at  a  high  altitude 
near  the  zenith.  At  the  foot  of  the  page  is  given  a  small 
supplementary  correction  depending  on  the  date  in  the  year. 
1  Which  leaves  refraction  only. 


THE   SEXTANT  73 

This  small  correction  is  not  important  in  navigation,  but  is 
given  here  for  the  sake  of  completeness.  It  arises  from  the 
semidiameter  part  of  the  combined  correction,  for  the  an- 
nual orbit  of  the  earth  around  the  sun  is  of  such  a  shape 
that  the  earth  is  nearer  the  sun  in  January  than  it  is  in  July, 
which  makes  the  sun  appear  bigger  in  January.  And  when  the 
sun  appears  big,  the  semidiameter  will  of  course  be  large  too. 

Table  7  gives  the  dip  of  the  sea  horizon,  the  number  in  the 
left-hand  column  being  the  height  (in  feet)  of  the  navigator's 
eye  above  sea  level.  This  will  be  the  height  of  the  ship's 
deck,  increased  by  the  height  of  the  man's  eye  above  the 
deck.  Unfortunately,  the  dip,  as  given  in  Table  7,  at  times 
varies  considerably  from  the  dip  as  it  actually  exists  at  the 
ship.  The  cause  can  be  seen  from  Fig.  17  (p.  70),  where 
it  will  be  noticed  that  the  line  from  the  observer  at  0  to  the 
sea  horizon  at  T'  passes  very  near  the  surface  of  the  ocean. 
It  is  therefore  entirely  in  the  lowest  strata  of  the  terrestrial 
atmosphere,  and  there  quite  irregular  refractions  sometimes 
occur.  These  have  been  known  to  produce  errors  in  the  dip 
amounting  to  10'  or  20',  and  it  is  principally  the  existence 
of  these  unavoidable  errors  that  makes  it  unnecessary  to 
read  the  sextant  closer  than  the  nearest  minute  (p.  66), 
when  observing  from  the  deck.  But  when  observing  ashore 
with  the  artificial  horizon,  which  has  no  dip,  the  navigator 
may,  if  he  chooses,  read  seconds,  especially  if  he  intends  to 
use  in  his  further  calculations  the  ''mean"  or  average  of 
a  considerable  number  of  observations. 

We  shall  now  give  an  example  of  the  complete  correction 
of  a  sextant  observation.  Suppose  the  angle  read  from 
the  sextant  was  30°  28',  the  index  error  (p.  68)  1',  addi- 
tive, height  of  observer's  eye  26  feet.  We  should  then 
have : 

observed  altitude,  lower  limb  =  30°  28' 

index  correction  =     +    1' 

correction  from  Table  6  (p.  247)  =     +  14' 

correction  from  Table  7  (p.  247)  =     —    5' 

corrected  altitude,  for  further  use        =  30°  38' 


74  NAVIGATION 

If  the  altitude  had  been  observed  ashore  with  an  arti- 
ficial horizon,  it  might  have  been  desirable  to  retain  seconds. 
The  calculation  might  then  have  been  as  follows : 

observed  double  altitude  (see  p.  70),  lower  limb     =  63°    0'  20" 

index  correction  (p.  68) 

corrected  double  altitude 

resulting  altitude 

correction  from  Table  6  (interpolated) 

corrected  altitude,  for  further  use 


+  1 

13 

63   1 

33 

31  30 

46 

+  14 

31 

31  45 

17 

CHAPTER  VII 
THE  NAUTICAL  ALMANAC 

Before  beginning  the  further  utiHzation  of  altitude  ob- 
servations in  our  navigation  calculations,  it  is  necessary  to 
understand  the  use  of  the  Nautical  Almanac.  This  is  an 
annual  publication,  issued  in  two  different  editions  by  the 
Nautical  Almanac  Office,  United  States  Naval  Observatory. 
Copies  can  be  obtained  from  the  Superintendent  of  Docu- 
ments, Washington,  D.  C,  or  through  any  dealer  in  nautical 
supplies.  Navigators  do  not  need  the  larger  edition,  of  which 
the  title  is  ''American  Ephemeris  and  Nautical  Almanac"; 
accordingly,  all  our  references  are  made  to  the  smaller  edi- 
tion for  the  year  1917.  Parts  of  certain  pages  from  that 
edition  are  reprinted  in  the  present  volume  for  convenience 
of  reference,  and  we  shall  give  a  somewhat  detailed  explana- 
tion of  the  almanac  page  29  (our  p.  76). 

Let  us  consider  the  date  Monday,  Dec.  17.  We  find  for 
that  date,  and  for  every  even  hour  (0'*,  2^*,  4^,  Q^,  etc.)  of 
"Greenwich  Mean  Time"  (abbreviated  G.  M.  T.^),  two 
tabular  numbers  (p.  10)  called  ''sun's  declination"  and 
"equation  of  time." 

To  understand  these  it  is  necessary  to  bear  in  mind  that 
the  kind  of  time  in  ordinary  use  is  "solar  time,"  as  kept  by 
the  sun.  The  "solar  day"  begins  at  "noon,"  called  0'*  in 
astronomic  navigation,  and  it  continues  through  twenty-four 
hours,  without  any  confusing  a.m.  and  p.m.  In  ordinary 
life  the  day  begins  twelve  hours  sooner,  at  midnight,  and 
runs  through  two  twelve-hour  periods  of  a.m.  and  p.m.  to 

1  The  reader  is  requested  to  note  carefully  this  abbreviation,  as 
it  will  be  used  very  frequently. 

75 


76 


NAVIGATION 


SUN,    DECEMBER,    1917.     From  Nautical  Almanac,  p.  29 


G.  M.  T. 

Sun's  Dec- 

Equation 

Sun's  Dec- 

Equation 

Sun's 

Dec- 

Equation 

lination 

OF  Time 

lination 

OF  Time 

LINATION 

OF  Time 

Monday  17 

Tuesday  25 

Saturday  29 

h 

o                / 

m     s 

o             / 

m       s 

o 

/ 

m     s 

0 

-  23  21.3 

+  3  56.8 

-  23  24.7 

-0     1.6 

-  23 

15.2 

-  1  59.7 

2 

23  21.5 

3  54.4 

23  24.6 

0     4.1 

23 

14.9 

2     2.1 

4 

23  21.7 

3  51.9 

23  24.5 

0     6.5 

23 

14.6 

2     4.6 

6 

23  21.9 

3  49.5 

23  24.4 

0     9.0 

23 

14.3 

2     7.0 

8 

23  22.1 

3  47.0 

23  24.2 

0  11.5 

23 

14.0 

2     9.4 

10 

23  22.2 

3  44.5 

23  24.1 

0  14.0 

23 

13.7 

2  11.9 

12 

23  22.4 

3  42.1 

23  24.0 

0  16.5 

23 

13.4 

2   14.3 

14 

23  22.6 

3  39.6 

23  23.8 

0  18.9 

23 

13.1 

2  16.7 

16 

23  22.8 

3  37.1 

23  23.7 

0  21.4 

23 

12.8 

2   19.1 

18 

23  22.9 

3  34.7 

23  23.5 

0  23.9 

23 

12.5 

2  21.5 

20 

23  23.1 

3  32.2 

23  23.4 

0  26.4 

23 

12.2 

2  24.0 

22 

23  23.2 

3  29.8 

23  23.2 

0  28.8 

23 

11.9 

2  26.4 

H.  D. 

0.1 

1.2 

0.1 

1.2 

0.1 

1.2 

Tuesday  18 

Wednesday  26 

Sunday  30 

0 

-  23  23.4 

+  3  27.3 

-  23  23.1 

-0  31.3 

-23 

11.6 

-  2  28.8 

2 

23  23.6 

3  24.8 

23  22.9 

0  33.8 

23 

11.3 

2  31.2 

4 

23  23.7 

3  22.3 

23  22.7 

0  36.3 

23 

11.0 

2  33.6 

6 

23  23.8 

3  19.9 

23  22.5 

0  38.7 

23 

10.6 

2  36.0 

8 

23  24.0 

3   17.4 

23  22.4 

0  41.2 

23 

10.3 

2  38.4 

10 

23  24.1 

3  14.9 

23  22.2 

0  43.7 

23 

10.0 

2  40.9 

12 

23  24.3 

.     3   12.5 

23  22.0 

0  46.2 

23 

9.7 

2  43.3 

14 

23  24.4 

3  10.0 

23  21.8 

0  48.6 

23 

9.3 

2  45.7 

16 

23  24.5 

3     7.5 

23  21.7 

0  51.1 

23 

9.0 

2  48.1 

18 

23  24.6 

3     5.0 

23  21.5 

0  53.6 

23 

8.6 

2  50.5 

20 

23  24.8 

3     2.6 

23  21.3 

0  56.0 

23 

8.3 

2  52.9 

22 

23  24.9 

3     0.1 

23  21.1 

0  58.5 

23 

7.9 

2  55.3 

H.  D. 

0.1 

1.2 

0.1 

1.2 

0.2 

1.2 

Wednesday  19 

Thursday  27 

Monday  31 

0 

-  23  25.0 

+  2  57.6 

-  23  20.9 

-  1     0.9 

-  23 

7.6 

-  2  57.7 

2 

23  25.1 

2  55.1 

23  20.7 

1     3.4 

23 

7.2 

3     0.1 

4 

23  25.2 

2  52.6 

23  20.5 

1     5.9 

23 

6.9 

3     2.4 

6 

23  25.3 

2  50.2 

23  20.3 

1     8.3 

23 

6.5 

3     4.8 

8 

23  25.4 

2  47.7 

23  20.1 

1   10.8 

23 

6.1 

3     7.2 

10 

23  25.5 

2  45.2 

23  19.8 

1   13.2 

23 

5.8 

3     9.6 

12       ' 

23  25.6 

2  42.7 

23  19.6 

1   15.7 

23 

5.4 

3   12.0 

14 

23  25.7 

2  40.2 

23   19.4 

1   18.1 

23 

5.0 

3  14.4 

16 

23  25.8 

2  37.8 

23  19.2 

1  20.6 

23 

4.6 

3  16.7 

18 

23  25.9 

2  35.3 

23  19.0 

1  23.1 

23 

4.3 

3  19.1 

20 

23  26.0 

2  32.8 

23  18.7 

1  25.5 

23 

3.9 

3  21.5 

22 

23  26.1 

2  30.3 

23  18.5 

1   28.0 

-  23 

3.5 

-  3  23.9 

H.  D. 

0.0 

1.2 

0.1 

1.2 

0.2 

1.2 

Thursday  20 

Friday  28 

0 

-  23  26.1 
23  26.2 

+  2  27.8 
2  25.3 

-  23  18.3 
23   18.0 

-  1   30.4 
1   32^9 

2 

4 

23  26.3 

2  22.8 

23  17.8 

1  35.3 

6 

23  26.3 

2  20.4 

23  17.5 

1  37.8 

8 

23  26.4 

2  17.9 

23  17.3 

1  40.2 

SEMIDIAMETER 

10 

23  26.5 
23  26.5 

2  15.4 
2  12.9 

23   17.0 
23  16.8 

1  42.6 
1  45.1 

12 

14 

23  26.6 

2  10.4 

23  16.5 

1  47.5 

Dec. 

1 

16'26 

16 

23  26.6 

2     7.9 

23  16.3 

1  50.0 

11 
21 
31 

16'28 
16'29 
16'30 

18 

23  26.7 

2     5.4 

23   16.0 

1   52.4 

20 

23  26.7 

2     2.9 

23  15.7 

1   54.8 

22 

-  23  26.8 

+  2     0.4 

-  23  15.4 

-  1  57.3 

H.  D. 

0.0 

1.2 

0.1 

1.2 

Note.  —  The  Equation  of  Time  ia  to  be  applied  to  the  G.  M.  T.  in  accordance  with 
the  sign  as  given. 


THE   NAUTICAL  ALMANAC  77 

the  following  midnight;  but  this  ''civil  day,"  as  it  is  called, 
does  not  for  the  moment  concern  us. 

Solal"  time,  as  kept  by  the  visible  sun,  is  a  very  incon- 
venient kind  of  time,  because  there  are  certain  peculiarities 
in  the  astronomic  motion  of  the  earth  which  make  these 
solar  days  of  unequal  length.  They  are  called  ''apparent 
solar  days"  and  the  corresponding  kind  of  time  is  "apparent 
solar  time." 

To  avoid  the  above  inconvenience,  an  imaginary  "mean 
sun"  and  a  "mean  solar  day"  have  been  invented.  The 
mean  sun  conforms  as  nearly  as  possible  to  the  average  per- 
formance of  the  visible  sun,  and  the  length  of  the  mean 
solar  day  is  the  average  of  all  the  apparent  solar  days  through- 
out the  year.  The  corresponding  kind  of  time,  kept  by  the 
mean  sun,  is  "mean  solar  time"  ;  and  this  is  the  kind  of  time 
recorded  by  all  our  w^atches  and  marine  chronometers  (p.  6). 

The  difference  between  these  two  kinds  of  solar  time  varies 
on  different  dates,  and  even  at  different  hours  on  the  same 
date.  It  is  this  difference  which  is  called  the  "equation  of 
time  "  and  which  is  one  of  the  tabular  numbers  in  the  nautical 
almanac  page  29  (our  p.  76). 

This  equation  of  time  is  of  great  importance  in  navigation, 
and  it  is  easy  to  see  how  page  29  of  the  almanac  may  be  used 
to  find  it.  Suppose,  for  instance,  we  wish  to  know  what  the 
equation  is  on  Dec.  17,  1917,  on  board  ship,  when  the  ship's 
chronometer  indicates  on  its  face  3  p.m.,  civil  time,  or  (which 
is  the  same  thing)  3^,  astronomical  time  (p.  75).  Ship's 
chronometers  are  always  set  to  Greenwich  mean  time,  so 
that  S^  by  the  chronometer  signifies  that  the  time  at  Green- 
wich was  3^. 

We  then  look  in  the  almanac  page  29  (our  p.  76),  and  find 
that  the  equation  was  +  S'"  54^4  at  2^,  G.  M.  T.,  and 
+  3"^  5P.9  at  #,  G.  M.  T.  Its  value  at  3^  must  be  half- 
way between  these  two,  or  +  3""  53*.  15.  This  we  would 
call  -f-  3""  53^2,  so  as  to  avoid  the  use  of  hundredths  of 
seconds,  which  do  not  need  attention  in  navigation.     And 


78  NAVIGATION 

since  the  equation  is  merely  the  difference  between  the 
two  kinds  of  solar  time,  the  +  sign  means  that  it  must  be 
added  to  G.  M.  T.,  to  obtain  Greenwich  apparent  time,  in 
accordance  with  the  ''Note"  at  the  foot  of  the  almanac 
page  29.  Consequently,  the  G.  M.  T.  by  chronometer  having 
been  S^  0""  0^  the  Greenwich  apparent  time  at  the  same  in- 
stant was  3^  0"^  0'  +  3"^  53'.2  =  3^  3^  53'.2. 

It  will  be  noticed  that  the  process  we  have  here  used  for 
obtaining  the  equation  from  the  almanac  is  merely  an  inter- 
polation (see  p.  12).  Let  us,  as  another  example,  find  the 
equation  for  Sunday,  Dec.  30,  at  10''  26""  a.m.,  civil  time  by 
chronometer,  and  we  have  purposely  here  retained  the 
civil  method  of  reckoning  time  to  make  certain  that  the 
reader  understands  the  difference  between  civil  and  astro- 
nomic (or  navigation)  time.  The  given  time  is  10^  26 ""  a.m., 
civil  time,  Dec.  30.  But  the  astronomic  Dec.  30  does  not 
begin  until  noon  (p.  75),  so  that  it  is  not  yet  Dec.  30  by 
astronomic  reckoning.  By  that  reckoning  it  is  really  only 
22^"  26*"  on  Dec.  29.  In  other  words,  when  the  civil  time  is 
P.M.,  as  in  the  first  example,  the  astronomic  time  is  the  same 
as  the  civil  time.  But  when  the  civil  time  is  a.m.,  as  in  the 
present  example,  the  astronomic  time  is  found  by  adding 
12^"  to  the  civil  time,  and  deducting  1  from  the  date.  These 
complications  emphasize  the  advantage  of  the  astronomic 
count,  which  avoids  a.m.  and  p.m.  altogether. 

We  now  have  from  the  almanac  (p.  76) : 

equation  of  time,  Dec.  29,  22^  G.  M.  T.  =  -  2"»  26^4, 
equation  of  time,  Dec.  30,     0\  G.  M:  T.  =  -  2*"  28^8 ; 

and  the  numbers  in  this  example  have  been  purposely  so 
chosen  that  the  above  two  tabular  values  of  the  equation 
(between  which  the  required  value  falls)  come  from  different 
dates  in  the  almanac.  This  creates  no  confusion,  for  these 
two  values  of  the  equation  are  really  consecutive  tabular 
numbers,  just  as  much  as  if  they  occurred  on  a  single  date. 
The  difference  between  the  two  values  of  the  equation  is 


THE   NAUTICAL  ALMANAC  79 

2* A;  and  as  this  difference  corresponds  to  2^  in  the  left' 
hand  (or  argument)  column,  it  follows  that  the  difference 
for  I''  is  here  P. 2.  This  is  the  change  of  the  equation  per 
hour  of  time;  it  is  called  the  ''hourly  difference"  (abbre- 
viated  H.  D.)  and  is  printed  in  the  almanac  at  the  foot  of 
each  daily  column. 

Now  we  want  the  equation  for  Dec.  29,  22^^  26"*,  by  the 
chronometer.  The  26""  must  next  be  changed  into  a  decimal 
fraction  of  an  hour.  26^  =  ff  of  an  hour  =  0''.43.  So  the 
time  for  which  we  want  the  equation  becomes  Dec.  29, 
22\43.  The  H.  D.  being  1^2,  the  change  in  0\43  will  be 
P.2  X  0.43  =  0^5.  The  almanac  shows  that  at  22^^  the  equa- 
tion was  2"^  26*. 4,  and  was  increasing  numerically.  There- 
fore, at  22^43,  it  was  2^  26^4  +  0^5  =  2^  26^9.  And  this 
number  has  the  minus  sign.  Therefore,  the  G.  M.  T.  being 
Dec.  29,  22^"  26"*,  the  Greenwich  apparent  time  at  the  same 
instant  will  be  Dec.  29,  22^*  26"*  -  2"*  26^9  =  Dec.  29, 
22^*  23"*  33M. 

Most  of  these  minor  interpolation  calculations,  which  are 
here  set  forth  in  great  detail  for  the  benefit  of  the  beginner, 
can  be  made  with  sufficient  accuracy  by  a  skilled  navigator 
mentally. 

In  the  foregoing  two  examples  we  have  assumed  that  the 
chronometer  was  right,  but  these  instruments  practically 
never  run  quite  correctly.  Therefore,  before  leaving  port, 
navigators  always  have  their  chronometers  ''rated"  by  a 
chronometer  expert ;  and  when  the  instrument  is  returned 
to  the  ship  just  before  sailing,  a  "rate  card"  (or  "rate  paper") 
always  comes  with  it.  Let  us  suppose  that  in  the  present 
example  this  card  stated  that  the  chronometer  was  slow 
8"*  22^5  1  on  Dec.  20,  at  noon,  and  was  "losing"  ^  p.8  daily. 
The  8"*  22^5  would  then  be  the  "chronometer  error"  on 
Dec.  20 ;  and  the  1'.8  would  be  its  "daily  rate." 

1  This  number  is  here  purposely  chosen  much  larger  than  would 
ever  occur  in  practice. 

2  The  opposite  kind  of  "rate"  is  called  "gaining." 


80  NAVIGATION 

From  Dec.  20,  noon,  to  Dec.  30,  lO'^  26"^  a.m.  is  an  interval 
of  9  days  22  hours  26  minutes.  This  interval  must  now  be 
reduced  to  a  decimal  of  a  day.  26""  =  f f  of  an  hour  =  0''.43. 
The  interval  is  therefore  9^^  22\43. 

But  22^43  =2.|^days  =  0*^.93.  Therefore,  in  days,  the 
interval  is  9*^.93.  This  transformation  of  hours  and  minutes 
into  decimals  of  a  day  can  be  accomplished  with  less  trouble  by 
means  of  our  Table  8  (p.  248). 

Having  a  losing  rate  of  P. 8  daily,  the  chronometer  lost 
P.8  X  9.93  =  17^9  in  the  interval  of  9.93  days.  And  as  it  was 
already  slow  8^  22^5  on  Dec.  20,  it  was  slow  8^  22^5  +  17^9 
=  8""  40^.4  at  the  time  for  which  the  equation  is  required. 

Now  the  equation  was  required  for  Dec.  29,  22^^  26*"  by  the 
chronometer ;  and  that  instrument  being  slow  8""  40'.4,  the 
correct  G.  M.  T.  was  :  Dec.  29,  22^^  26^  +  8"^  40^4  =  Dec.  29, 
22^"  34""  40^4.  Turned  into  a  decimal  fraction  of  an  hour, 
this  becomes  Dec.  29,  22^*. 58,  instead  of  22''.43,  as  we  found 
before,  when  the  chronometer  error  was  omitted  from  the 
calculation.  The  H.  D.  is  1*.2,  as  before,  and  the  change 
in  0\58  =  P.2  X  0.58  =  0«.7.  Therefore,  at  22^58  the 
equation  is  2"»  26^4  +  0\7  =  2^  27M.  This  still  has  the 
minus  sign,  so  that  the  correct  Greenwich  apparent  time 
becomes  Dec.  29,  22^^  34^  40«.4  -  2^  27M  =  22^^  32^  13«.3. 

All  the  above  calculations  have  been  carried  out  here  with 
unnecessary  accuracy.  There  would  be  no  harm  if  the  result 
were  in  error  by  a  few  tenths  of  a  second ;  and  it  is  this  cir- 
cumstance that  makes  it  possible  to  perform  these  inter- 
polations largely  mentally. 

In  the  foregoing  examples  no  account  was  taken  of  the 
ship's  location  on  the  ocean ;  yet  this  location  may  have  an 
indirect  influence  on  the  calculations.  To  understand  this, 
we  must  consider  for  a  moment  the  time-differences  which 
exist  between  different  places  on  the  earth.  The  sun  rises  in 
the  east  and  travels  across  the  sky  toward  the  west ;  so  that 
if  we  consider  two  places  like  Greenwich,  England,  and  New 
York,  for  instance,  the  sun,  because  of  this  motion  from  east 


THE   NAUTICAL  ALMANAC  81 

to  west,  will  pass  Greenwich  first.     Consequently,  when  it  is 

noon  in  New  York,  it  has  already  been  noon  in  Greenwich, 

and  is  afternoon  there.     Greenwich  time  is  therefore  always 

later  than  New  York  time.     The  same  is  true  of  any  other 

two  places  ;  there  is  always  a  time-difference  between  them, 

and  the  easterly  place  has  the  later  or  '^ faster"  time. 

The  amount  of  such  time-difference  of  course  depends 

on  the  relative  location  of  the  two  places,  and  the  relation  is 

such  that    15°  of  longitude-difference   corresponds  exactly 

to   I'*  of  time-difference.     Thus  Sandy  Hook,  which  is  in 

longitude  73°  50'  west  of  Greenwich,  has  a  time-difference 

from  Greenwich  of  4^  55""  20^     This  conversion  of  longitude 

into  time-difference  is  best  accomplished  by  means  of  our 

Table  9  (p.  249).     According  to  that  table : 

73°  =  4^  52«    0* 
50'  3     20 

73°  50'  =  4^  55^  20* 

The  indirect  influence  of  such  time-differences  upon  the 
use  of  the  almanac  is  that  they  ma}'  at  times,  especially 
when  they  are  large,  make  the  Greenwich  date  of  the  ob- 
servation different  from  the  date  on  board.  Thus  a  vessel 
off  Manila  Bay,  in  longitude  120°  east  of  Greenwich,  would 
have  her  local  time  8^  (120°)  later  than  Greenwich  time.  If 
a  sextant  observation  was  made  on  board  at  4  p.m.,  civil 
time,  on  a  Thursday,  the  chronometer  would  indicate  8^*, 
and  it  would  be  8  a.m.  on  Thursdav,  because  Greenwich  is 
S^  earlier  than  the  ship.  This  8  a.m.  would  really  be  20^  of 
the  preceding  Wednesday  by  astronomic  time,  and  so  the 
almanac  date  used  would  be  one  day  earher  than  the  date 
of  the  observation.  The  chronometer  will  always  give  the 
right  Greenwich  time,  but  the  navigator  must  be  very  care- 
ful to  interpolate  the  almanac  numbers  on  the  right  date. 

We  have  now  learned  how  to  ascertain  the  equation  of 
time  from  the  almanac,  and  how  to  use  it  for  transforming 
G.  M.  T.  into  Greenwich  apparent  time.  The  contrary 
transformation,  from  Greenwich  apparent  time  to  G.  M.  T., 


82  NAVIGATION 

can  be  made  by  applying  the  equation  in  the  opposite  way : 
subtracting  when  it  has  the  +  sign  in  the  almanac,  and  add- 
ing when  it  has  the  —  sign. 

The  great  importance  of  these  time  transformations  comes 
from  the  fact  that  sextant  observations  must  necessarily  be 
made  upon  the  visible  sun.  When  they  are  made  for  the 
purpose  of  calculating  the  local  time  on  board,  this  local 
time  will  therefore  necessarily  be  local  apparent  solar  time,  as 
kept  by  the  visible  sun.  At  the  instant  of  the  observation 
(p.  6),  the  chronometer  face  (corrected  for  error  and  rate) 
tells  us  the  G.  M.  T.  If  this  is  turned  into  Greenwich  ap- 
parent time  by  applying  the  equation,  we  have  only  to  com- 
pare the  Greenwich  and  the  ship's  apparent  times  to  get 
the  time-difference  between  the  ship  and  Greenwich.  This 
time-difference  can  then  be  turned  into  degrees  and  minutes, 
and  will  be  the  ship's  longitude.  Examples  of  this  calcu- 
lation will  be  given  in  detail  (p.  99).  It  is  also  worth 
noting  here  that  the  time-difference  between  any  two  places 
is  precisely  the  same,  quite  irrespective  of  the  kind  of  time 
in  which  it  is  counted. 

To  complete  our  explanation  of  the  almanac  page  29  (our 

p.  76),  it  remains  to  give  an  example  of  a  calculation  of  the 

sun's  declination.     This  is  an  angle  in  degrees  and  minutes, 

and  it  is  interpolated  just  like  the  equation  by  the  aid  of 

its  H.  D.     Thus,  for  Dec.  29,  22\58  (p.  80)  the  dechnation 

is  obtained  thus : 

Dec.  29,  22\  declination  =  23°  11  '.9 

H.D.  (OM)  X  O^'.SS  =  0.1,  declination  decreasing ; 

by  subtraction,  at  22^58,  dec.  =  23°  11 '.8, 

and  according  to  the  almanac,  this  declination  must  be  given 
the  minus  sign.  When  the  sign  should  be  +,  that  fact  is 
indicated  in  the  almanac.  The  use  of  the  declination  will 
be  explained  later;  the  accuracy  required  in  the  interpo- 
lation of  it  is  not  so  great  as  we  have  used  here,  for  the 
nearest  minute  suffices  in  practically  all  navigation  work. 
In  addition  to  the  sun's  declination,  navigators  require 


THE  NAUTICAL  ALMANAC 


83 


in  their  further  calculations  another  number  called  the  sun's 
"right  ascension"  (abbreviated,  R.  A.).  This  is  obtained 
from  pages  like  the  almanac  page  3  (reprinted  in  part  below) . 
It  is  always  the  R.  A.  of  the  "mean  sun"  that  we  need, 
and  the  almanac  gives  it  for  Greenwich  mean  noon  of  each 
day  in  the  year.  When  needed  in  our  further  calcula- 
tions, it  is  of  course  always  required  for  the  exact  moment 
when  a  sextant  observation  was  made.  In  fact,  this  state- 
ment applies  also  to  the  equation  of  time  and  declination. 
They  must  always  be  interpolated  from  the  almanac  for  the 
moment  when  the  navigator  actually  observed  the  sun ;  and 

SUN,    1917.     From  Nautical  Almanac,  p.  3 


Day 

Right  . 

A.SCENSION  OF  THE 

VIean 

Sun  at  Greenwich  Mean 

Noon 

OF 

Month 

July 

August 

September 

October 

November 

December 

h 

m 

s 

h 

m 

s 

h 

m 

s 

h 

m 

s 

h 

m 

s 

h 

m   a 

1 

6 

35 

52.2 

8 

38 

5.5 

10 

40 

18.7 

12 

38 

35.3 

14 

40 

48.4 

16 

39  5.1 

2 

6 

39 

48.8 

8 

42 

2.0 

10 

44 

15.2 

12 

42 

31.8 

14 

44 

45.0 

16 

43  1.7 

3 

6 

43 

45.3 

8 

45 

58.6 

10 

48 

11.8 

12 

46 

28.4 

14 

48 

41.5 

16 

46  58.2 

4 

6 

47 

41.9 

8 

49 

55.1 

10 

52 

8.3 

12 

50 

24.9 

14 

52 

38.1 

16 

50  54.8 

5 

6 

51 

38.4 

8 

53 

51.7 

10 

56 

4.9 

12 

54 

21.5 

14 

56 

34.6 

16 

54  51.3 

6 

6 

55 

35.0 

8 

57 

48.2 

11 

0 

1.4 

12 

58 

18.0 

15 

0 

31.2 

16 

58  47.9 

7 

6 

59 

31.6 

9 

1 

44.8 

11 

3 

58.0 

13 

2 

14.6 

15 

4 

27.8 

17 

2  44.5 

8 

7 

3 

28.1 

9 

5 

41.4 

11 

7 

54.5 

13 

6 

11.1 

15 

8 

24.3 

17 

6  41.0 

9 

7 

7 

24.7 

9 

9 

37.9 

11 

11 

51.1 

13 

10 

7.7 

15 

12 

20.9 

17 

10  37.6 

10 

7 

11 

21.2 

9 

13 

34.5 

11 

15 

47.6 

13 

14 

4.2 

15 

16 

17.4 

17 

14  34.1 

11 

7 

15 

17.8 

9 

17 

31.0 

11 

19 

44.2 

13 

18 

0.8 

15 

20 

14.0 

17 

18  30.7 

12 

7 

19 

14.3 

9 

21 

27.6 

11 

23 

40.8 

13 

21 

57.3 

15 

24 

10.5 

17 

22  27.2 

13 

7 

23 

10.9 

9 

25 

24.1 

11 

27 

37.3 

13 

25 

53.9 

15 

28 

7.1 

17 

26  23.8 

14 

7 

27 

7.4 

9 

29 

20.7 

11 

31 

33.9 

13 

29 

50.4 

15 

32 

3.6 

17 

30  20.4 

15 

7 

31 

4.0 

9 

33 

17.2 

11 

35 

30.4 

13 

33 

47.0 

15 

36 

0.2 

17 

34  16.9 

16 

7 

35 

0.6 

9 

37 

13.8 

11 

39 

27.0 

13 

37 

43.6 

15 

39 

56.8 

17 

38  13.5 

17 

7 

38 

57.1 

9 

41 

10.4 

11 

43 

23.5 

13 

41 

40.1 

15 

43 

53.3 

17 

42  10.0 

18 

7 

42 

53.7 

9 

45 

6.9 

11 

47 

20.1 

13 

45 

36.7 

15 

47 

49.9 

17 

46  6.6 

19 

7 

46 

50.2 

9 

49 

3.5 

11 

51 

16.6 

13 

49 

33.2 

15 

51 

46.4 

17 

50  3.2 

20 

7 

50 

46.8 

9 

53 

0.0 

11 

55 

13.2 

13 

53 

29.8 

15 

55 

43.0 

17 

53  59.7 

21 

7 

54 

43.4 

9 

56 

56.6 

11 

59 

9.7 

13 

57 

26.3 

15 

59 

39.5 

17 

57  56.3 

22 

7 

58 

39.9 

10 

0 

53.1 

12 

3 

6.3 

14 

1 

22.9 

16 

3 

36.1 

18 

1  52.8 

23 

8 

2 

36.5 

10 

4 

49.7 

12 

7 

2.8 

14 

5 

19.4 

16 

7 

32.6 

18 

5  49.4 

24 

8 

6 

33.0 

10 

8 

46.2 

12 

10 

59.4 

14 

9 

16.0 

16 

11 

29.2 

18 

9  46.0 

25 

8 

10 

29.6 

10 

12 

42.8 

12 

14 

55.9 

14 

13 

12.5 

16 

15 

25.8 

18 

13  42.5 

26 

8 

14 

26.1 

10 

16 

39.4 

12 

18 

52.5 

14 

17 

9.1 

16 

19 

22.3 

18 

17  39.1 

27 

8 

18 

22.7 

10 

20 

35.9 

12 

22 

49.0 

14 

21 

5.6 

16 

23 

18.9 

18 

21  35.6 

28 

8 

22 

19.2 

10 

24 

32.4 

12 

26 

45.6 

14 

25 

2.2 

16 

27 

15.4 

18 

25  32.2 

29 

8 

26 

15.8 

10 

28 

29.0 

12 

30 

42.2 

14 

28 

58.8 

16 

31 

12.0 

18 

29  28.7 

30 

8 

30 

12.4 

10 

32 

25.6 

12 

34 

38.7 

14 

32 

55.3 

16 

35 

8.6 

18 

33  25.3 

31 

8 

34 

8.9 

10 

36 

22.1 

12 

38 

35.3 

14 

36 

51.9 

16 

39 

5.1 

18 

37  21.9 

84 


NAVIGATION 


CORRECTION  TO  BE  ADDED  TO  R.  A.  M.  S.  AT  G.  M.    N.    FOR 

TIME    PAST    NOON 

From  Nautical  Almanac,  p.  3,  Continued 


Time 

Qm 

6« 

12'^ 

IS"* 

24m 

ZQrn 

36'« 

42m 

48'" 

Time 

h 

m  s 

m  s 

m   s 

m 

s 

m 

8 

m 

s 

m   3 

m 

s 

m  8 

h 

12 

1  58.3 

1  59.3 

2  0.2 

2 

1.2 

2 

2.2 

2 

3.2 

2  4.2 

2 

5.2 

2  6.2 

12 

13 

2  8.1 

2  9.1 

2  10.1 

2 

11.1 

2 

12.1 

2 

13.1 

2  14.0 

2 

15.0 

2  16.0 

13 

14 

2  18.0 

2  19.0 

2  20.0 

2 

20.9 

2 

21.9 

2 

22.9 

2  23.9 

2 

24.9 

2  25.9 

14 

15 

2  27.8 

2  28.8 

2  29.8 

2 

30.8 

2 

31.8 

2 

32.8 

2  33.8 

2 

34.7 

2  35.7 

15 

16 

2  37.7 

2  38.7 

2  39.7 

2 

40.7 

2 

41.6 

2 

42.6 

2  43.6 

2 

44.6 

2  45.6 

16 

17 

2  47.6 

2  48.5 

2  49.5 

2 

50.5 

2 

51.5 

2 

52.5 

2  53.5 

2 

54.5 

2  55.4 

17 

18 

2  57.4 

2  58.4 

2  59.4 

3 

0.4 

3 

1.4 

3 

2.3 

3  3.3 

3 

4.3 

3  5.3 

18 

19 

3  7.3 

3  8.3 

3  9.2 

3 

10.2 

3 

11.2 

3 

12.2 

3  13.2 

3 

14.2 

3  15.2 

19 

20 

3  17.1 

3  18.1 

3  19.1 

3 

20.1 

3 

21.1 

3 

22.1 

3  23.0 

3 

24.0 

3  25.0 

20 

21 

3  27.0 

3  28.0 

3  29.0 

3 

29.9 

3 

30.9 

3 

31.9 

3  32.9 

3 

33.9 

3  34.9 

21 

22 

3  36.8 

3  37.8 

3  38.8 

3 

39.8 

3 

40.8 

3 

41.8 

3  42.8 

3 

43.7 

3  44.7 

22 

23 

3  46.7 

3  47.7 

3  48.7 

3 

49.7 

3 

50.6 

3 

51.6 

3  52.6 

3 

53.6 

3  54.6 

23 

the  Greenwich  time  of  this  event  is  of  course  always  taken 
from  the  chronometer  (duly  corrected  for  error  and  rate). 

Thus,  if  the  R.  A.  of  the  mean  sun  is  required  for  Dec.  29, 
22*  34"*  40'.4,  G.  M.  T.  (p.  80),  we  find  from  the  almanac 
page  3  (our  p.  83)  that  the  R.  A.  of  the  mean  sun  at  Green- 
wich mean  noon  is  18*  29"*  28*.7.^  This,  according  to  the  sup- 
plementary table  quoted  above  from  page  3,  must  be  increased 
by  a  correction  for  ''time  past  noon."  In  this  case  the  time 
past  noon  is  22*  34*"  40'.4.  The  tabular  correction  for  22*  30"* 
is  3**  4P.8,  and  for  22*  36"*  it  is  3"»  42».8.  Ours  falls  between 
these  two,  and  an  interpolation  makes  the  correction  3"*  42*. 6. 
Consequently,  the  R.  A.  of  the  mean  sun  for  Dec.  29,  22* 
34m  4o«.4^  G.  M.  T.  is  18*  29"*  28*.7  +  3"*  42'.6  =  18*  33"*  1P.3. 

It  will  be  noticed  that  the  small  supplementary  table 
(quoted  above  from  almanac  page  3)  only  runs  from  12*  to  24*. 
The  other  half  of  the  table,  from  0*  to  12*,  is  printed  on  the 
opposite  page  2  of  the  almanac.  There  is  also  another 
longer  table,  printed  near  the  end  of  the  almanac,  and  there 
called  Table  III,  from  which  the  supplementary  correction 
can  be  taken  without  the  necessity  of  interpolation. 

It  is  not  absolutely  essential  that  the  navigator  learn  what 

^  Right  ascensions  are  always  thus  measured  in  hours,  minutes, 
and  seconds,  like  time,  and  they  are  counted  from  0*  to  24''. 


THE  NAUTICAL  ALMANAC  85 

the  words  ''right  ascension"  and  ''  decHnation"  really  mean. 
But  for  the  benefit  of  those  who  are  curious  in  such  matters 
we  may  state  that  these  numbers  locate  the  position  of  the 
sun  (or  of  a  star)  on  the  sky.  The  sky  is  a  great  globe,  called 
by  astronomers  the  ''celestial  sphere,"  and  all  heavenly 
bodies  are  located  upon  it  precisely  as  points  on  the  earth 
are  there  located  by  their  latitudes  and  longitudes  (p.  3). 
There  is  a  "celestial  equator"  with  two  "celestial  poles," 
corresponding  accurately  to  the  terrestrial  equator  and  poles. 
Declination  then  corresponds  exactly  to  latitude  on  the  earth, 
and  so  it  measures  the  distance  of  a  heavenly  body  from  the 
celestial  equator.  When  the  body  is  north  of  the  celestial 
equator,  the  declination  is  called  +. 

Right  ascension  similarly  corresponds  to  longitude  ;  and  for 
the  beginning  point  of  right  ascensions  on  the  sky  there  is  a 
"celestial  Greenwich,"  which  is  called  the  "vernal  equinox." 

After  this  brief  digression  into  astronomy,  we  return  to 
our  subject.  We  have  seen  (p.  82)  that  observations  of 
the  sun  will  tell  us  only  apparent  solar  time,  because  it  is 
only  the  visible  sun  that  we  can  observe.  If  the  observations 
are  made  upon  a  star,  the  kind  of  time  is  different  from  any 
so  far  mentioned.     It  is  called  "sidereal  time,"  or  star  time. 

It  is  always  possible  to  change  mean  solar  time  into  sidereal 
time,  and  vice  versa,  by  a  simple  process  of  calculation ;  but 
the  only  change  of  this  kind  required  in  navigation  is  the 
transformation  of  G.  M.  T.  into  Greenwich  sidereal  time. 
To  make  this  transformation,  we  have  only  to  take  from  the 
almanac,  for  the  given  G.  M.  T.,  the  R.  A.  of  the  mean  sun, 
and  then  to  add  it  to  the  given  G.  M.  T. 

Thus,  to  find  the  Greenwich  sidereal  time  corresponding 
to  Dec.  29,  22*  34'"  40^4,  G.  M.  T.,  we  have  already  found 
(p.  84)  that  the  R.  A.  of  the  mean  sun  =  18^^33'"  11'.3 

To  this  must  be  added  the  given  G.  M.  T.     =  22  34   40.4 
Sum=  corresponding  Greenwich  sidereal  time  =  17^^  7"*  51*. 7 

^  The  number  of  hours  was  here  really  41* :  but  whenever  it  is 
larger  than  24^^,  we  must  drop  or  reject  24*. 


CHAPTER  VIII 
OLDER  NAVIGATION   METHODS 

We  shall  now  explain  in  detail  certain  standard  methods 
of  determining  a  ship's  latitude  and  longitude  by  means  of 
sextant  observations.  An  understanding  of  these  methods 
is  essential  to  a  proper  comprehension  of  the  newer  naviga- 
tional processes  to  be  described  later ;  and  the  older  methods 
are  in  fact  still  very  widely  used  at  sea,  although  most  re- 
cent authorities  believe  they  should  be  rejected  in  favor  of 
the  newer  procedure. 

The  simplest  of  these  older  processes,  and  the  one  most 
frequently  employed,  is  the  determination  of  the  ship's 
latitude  by  a  noon  or  '^meridian"  observation  (''noon- 
sight")  of  the  sun's  altitude  (p.  61).  Now  the  sun  is 
higher  in  the  sky  at  noon  than  it  is  at  any  other  time  during 
the  day;  and  so  it  is  possible  to  get  the  noon-sight  by  be- 
ginning to  observe  the  sun  with  the  sextant  a  few  minutes 
before  noon,  and  continuing  the  observation  as  long  as  the 
sun's  altitude  is  increasing.  The  moment  it  begins  to 
diminish,  or  the  sun  to  ''dip,"  as  sailors  say,  the  observation 
should  be  terminated,  and  the  vernier  read. 

The  altitude  thus  observed  will  be  an  altitude  of  the  lower 
limb  (p.  71) ;  and  before  it  is  used  further  it  must  be  fully 
corrected  for  index  error ;  for  refraction  parallax  and  semi- 
diameter  ;  and  for  dip ;  all  as  in  the  example  on  p.  73, 
where  the  observed  altitude  was  30°  28',  and  we  found  the 
corrected  altitude  to  be  30°  38'. 

Next,  the  sun's  declination  must  be  taken  from  the  al- 
manac, being  interpolated  for   the  Greenwich  time  of  the 

86 


OLDER   NAVIGATION  METHODS  87 

observation,  as  in  the  example  on  p.  82,  where  we  found 
the  dedination  to  be  -  23°  12'  on  Dec.  29,  at  22*  34'^  40'.4, 
G.  M.  T.  We  shall  suppose  the  above  altitude  30°  28'  to 
have  been  observed  at  the  Greenwich  time  stated,  so  as  to 
make  use  of  the  results  of  our  former  calculated  examples. 
Nor  is  there  any  inconsistency  in  supposing  a  noon  observa- 
tion to  have  been  made  at  22*  34*"  40*.4.  For  the  noon 
observation  is  made  when  it  is  noon  on  board  ship,  while 
the  22*  34"*  40'.4  is  the  G.  M.  T.  at  the  same  moment. 
The  difference  is  simply  the  time-difference  (p.  80)  between 
Greenwich  and  the  ship. 

The  calculation  of  the  ship's  latitude  is  now  made  by  the 
following  formula : 

Latitude  =  90°  +  Declination  —  Altitude. 

In  this  formula,  the  plus  sign  signifies  that  the  declination 
must  be  added;  and  the  minus  sign  signifies  that  the  altitude 
must  be  subtracted.  Furthermore,  it  is  most  important  to 
remember  that  if  the  declination  is  itself  a  "minus  declina- 
tion," as  in  this  example,  the  addition  of  it  according  to  the 
formula  is  really  a  subtraction.  Or,  in  other  words,  and  in 
general,  whenever  a  formula  calls  for  an  addition,  and  the 
number  to  be  added  is  a  minus  number,  then  that  number 
must  be  subtracted  instead  of  added.  And  similarly,  if  the 
formula  calls  for  a  subtraction,  and  the  number  to  be  sub- 
tracted is  a  minus  number,  then  that  number  must  be  added 
instead  of  subtracted.    Two  minus  signs  neutralize  each  other. 

In  the  present  case  we  have,  omitting  seconds : 

90°  0' 
declination  =  —  23  12 

90°  +  declination  =     66  48 
altitude  =     30  38 

latitude  =     36  10 

In  considering  this  result  it  is  of  interest  to  inquire  where 
this  observation  really  locates  the  ship.  Now  we  have  not 
yet  stated  what  the  date  was,  on  board,  when  the  observa- 


88  NAVIGATION 

tion  was  made;  but  we  have  given  the  G.  M.  T.  as  Dec.  29, 
22*  34"*  40'.4.  The  noon-sight  was  taken,  as  a  matter  of 
fact,  at  noon  on  Dec.  30,  or  at  the  moment  when  the  date 
Dec.  30  commenced  by  astronomic  reckoning.  Therefore 
the  ship's  time  was  later  than  the  Greenwich  time  by  about 
lA  25"*;  or  21°  15',  allowing  15°  to  1*  (p.  81) ;  and  the  ship 
was  (approximately)  in  21°  15'  east  longitude  from  Greenwich. 
This,  together  with  the  latitude  36°  10',  locates  the  ship  in 
the  Mediterranean,  south  of  Greece,  and  west  of  Candia. 

Although  we  have  thus  apparently  located  the  ship  com- 
pletely in  latitude  and  longitude  from  a  single  noon-sight, 
it  must  not  be  supposed  that  we  have  really  accomplished 
this.  The  noon-sight  is  only  suitable  for  ascertaining  the 
ship's  latitude ;  the  longitude  is  determined  so  inaccurately 
as  to  be  practically  useless.  The  reason  for  this  is  that 
near  noon  the  sun  changes  its  altitude  very  slowly,  because 
it  is  then  near  the  turning-point  where  its  upward  morning 
motion  is  about  to  become  a  downward  afternoon  motion. 
For  the  sun's  daily  motion  in  the  sky  is  upward  in  the  morn- 
ing and  downward  in  the  afternoon.  Near  noon  it  runs 
along  horizontally,  or  very  nearly  so,  for  several  minutes, 
so  that  its  altitude  change  is  insignificant  during  that  time. 

It  follows  from  this  temporary  invariability  of  altitude 
that  we  cannot  determine  the  exact  moment  when  noon 
occurs  by  observing  altitude  changes  with  the  sextant.  But 
the  latitude  determination  is  not  affected ;  because,  for 
the  latitude,  we  only  need  to  know  the  noon  altitude.  And 
if  we  happen  to  measure  it  a  little  too  soon  or  too  late,  on 
account  of  the  difficulty  of  fixing  the  moment  of  noon,  no 
harm  will  result,  because  the  altitude  very  near  noon  is  the 
same  as  it  is  at  noon  precisely,  as  we  have  just  seen. 

It  is,  in  general,  practically  impossible  to  determine  both 
latitude  and  longitude  from  a  single  observation.  To  deter- 
mine two  unknown  things,  at  least  two  different  observations 
must  be  made.  Nor  can  any  skillful  method  of  planning 
the  observation  overcome  this  fundamental  circumstance. 


OLDER  NAVIGATION  METHODS  89 

Returning  now  to  our  latitude  formula  (p.  87),  it  is 
necessary  to  modify  it  somewhat  in  case  we  happen  to  be  in 
the  tropics,  where  the  sun  may  pass  between  the  zenith  and 
the  celestial  pole.  Even  in  temperate  latitudes  a  celestial 
body  may  do  this,  if  we  happen  to  observe  a  star  instead  of 
the  sun.  In  such  a  case,  if  the  ship  is  in  the  northern 
hemisphere,  the  navigator  will  observe  the  sun's  altitude 
toward  the  north  at  noon  instead  of  toward  the  south,  as 
usual.  Furthermore,  in  very  high  northern  latitudes,  the 
''midnight  sun,"  as  it  is  called,  can  be  observed  toward  the 
north,  and  below  the  celestial  pole.  This  is  the  minimum 
altitude  during  the  day,  instead  of  the  maximum ;  but  it  is 
usable  for  a  latitude  determination.  Such  an  observation  is 
called  a  ''lower  transit"  ;  and  it  can  often  be  observed  in  the 
case  of  stars  in  temperate  latitudes. 

If  we  now  remember  to  call  northerly  latitudes  and 
declinations  plus,  and  southerly  ones  minus,  we  have  the 
following  complete  set  of  formulas  for  the  present  problem, 
including  observations  in  both  hemispheres.  These  formulas 
are  so  arranged  that  we  can  easily  choose  the  right  formula, 
by  having  regard  to  the  +  and  —  signs.  But  the  right 
formula  once  chosen,  the  latitude  is  calculated  without 
marking  declinations  with  either  the  +  or  —  sign. 

lat.i  and  (if  lat.  greater  than  dec,  lat.  =  90°  +  dec.  —  alt.  (1) 
dec.  both  +  j  if  dec.  greater  than  lat.,  lat.  =  dec.  +  alt.  —  90°  (2) 
or  both  —      [  if  lower  transit,  lat.  =  90°  +  alt.  —  dec.       (3) 

lat.  and  dec,  1  1^^  ^  9qo  _  ^j^  _  ^^^        ^^^ 

one  +,  one  —  J 

We  shall  now  give  some  more  examples ;    and  to  enable 

the  reader  to  follow  star  observations  correctly  we  reprint 

part  of  the  upper  halves  of  pages  94  and  95  (our  pp.  91,  92) 

of  the  Nautical  Almanac.     These  contain  the  right  ascensions 

and  declinations  (p.  85)  of  a  quantity  of  bright   stars  for 

various  dates  in  the  year.     These  numbers  are  correct  for  the 

moment  of  "upper  transit,"  which  is  the  moment  when  these 

1  Latitude  and  declination  are  abbreviated  lat.  and  dec. 


90  NAVIGATION 

stars  attain  their  maximum  altitudes.  This  event  cannot 
be  called  a  noon-sight  in  the  case  of  a  star ;  but  it  is  observable 
in  a  manner  perfectly  similar  to  a  solar  noon-sight. 

These  stellar  right  ascensions  and  declinations  change 
so  slowly  that  it  is  unnecessary  to  use  interpolation  when 
taking  them  from  the  almanac  pages. 

Proceeding  now  to  our  examples,  suppose  that  on  shore, 
at  Sandy  Hook  Light,  approximate  latitude  and  longitude 
40°  28'  N.,  74°  0'  W.,  on  Monday,  Dec.  17,  1917,  at  noon,  the 
double  altitude  of  the  sun's  lower  limb  was  observed  with  a 
sextant  and  artificial  horizon,  and  found  to  be  51°  48'.  The 
index  correction  required  by  the  sextant  was  -f  4' ;  and  the 
G.  M.  T.  by  chronometer  was  4^*  56"*  at  the  moment  the 
observation  was  made.     Find  the  latitude.     We  have : 

Observed  double  altitude 51°    48'  (1) 

Index  correction +4  (2) 

Adding  (1)  and  (2)  gives  corrected  double  altitude  . . .  .51°    52'  (3) 

Halving  (3)  gives  observed  altitude 25     56  (4) 

Correction  from  Table  6 1  (p.  247) + 14  (5) 

Adding  (4)  and  (5)  gives  fully  corrected  altitude 26°    10'  (6) 

Now  use  formula   (4)    (p.    89)    because   latitude  is  + 

and  declination  is  —  .    Write 90       0  (7) 

Subtracting  (6)  from  (7)  gives  90°  -  corrected  altitude . .  63  50  (8) 
Interpolate  declination  from  almanac  (p.   76).     This 

gives  declination 23     22  (9) 

Subtracting  (9)  from  (8)  gives  for  the  latitude 40     28  (10) 

With  regard  to  the  foregoing  example  it  is  worth  remark- 
ing that  if  there  had  been  no  available  chronometer  set  to 
Greenwich  time,  it  would  still  have  been  possible  to  calculate 
the  observation.  For  the  known  approximate  longitude, 
even  if  only  a  dead-reckoning  (p.  5)  longitude,  would  be 
quite  accurate  enough  to  make  possible  the  interpolation  of 
the  declination  from  the  almanac.  And  in  the  present 
example,  the  chronometer  was  only  used  in  getting  the 
declination  printed  in  line  (9)  above. 

1  Dip  correction  from  Table  7  not  needed  because  the  artificial 
horizon  was  used. 


OLDER  NAVIGATION  METHODS 


91 


APPARENT   PLACES   OF   STARS,    1917 

From  Nautical  Almanac,  p.  94 
For  the  Upper  Transit  at  Greenwich 


Constella- 

Right Ascension 

No. 

I-H 

i-H 

y-t 

,—1 

»-( 

i-< 

>-i 

1-H 

CO 

tion  Name 

d 

^ 

ii 

>, 

t« 

-I-' 

> 

6 

o 

•-» 

S 

i-s 

3 
S 

3 
< 

^ 

O 

^ 

Q 

Q 

h    m 

s 

s 

s 

8 

s 

s 

s 

s 

s 

1 

a  Androm. 

0    4 

6.3 

6.4 

7.4 

8.4 

9.4 

10.0 

10.3 

10.3 

10.0 

9.6 

2 

^  Cassiop. 

0    4 

44.8 

44.4 

45.7 

47.3 

48.7 

49.7 

50.1 

49.9 

49.3 

48.4 

3 

^Ceti 

0  39 

26.5 

26.3 

27.0 

28.0 

28.9 

29.7 

30.0 

30.1 

29.8 

29.5 

4 

5  Cassiop. 

1  20 

23.9 

22.3 

23.5 

25.1 

26.7 

28.1 

28.9 

29.2 

29.0 

28.2 

5 

aUrs.  Min. 

1  29 

89.0 

22.9 

45.5 

77.6 

112.8 

142.4 

161.2 

166.4 

155.3 

129.0 

6 

a  Eridani 

1  34 

39.1 

36.8 

37.6 

38.8 

40.3 

41.5 

42.3 

42.4 

41.9 

41.1 

7 

a  Arietis 

2    2 

31.0 

30.1 

30.8 

31.7 

32.7 

33.6 

34.3 

34.6 

34.7 

34.5 

8 

0  Eridani 

2  55 

8.8 

6.8 

7.2 

7.9 

9.0 

10.0 

10.8 

11.3 

11.4 

11.0 

9 

a  Persei 

3  18 

25.9 

23.9 

24.4 

25.5 

26.8 

28.2 

29.3 

30.2 

30.6 

30.5 

10 

a  Tauri 

4  31 

11.7 

10.3 

10.5 

11.0 

11.9 

12.8 

13.7 

14.5 

15.0 

15.2 

11 

/3  Orionis 

5  10 

35.1 

33.7 

33.7 

34.2 

34.7 

35.6 

36.5 

37.3 

37.8 

38.1 

12 

a  Aurigse 

5  10 

36.5 

34.5 

34.6 

35.2 

36.2 

37.5 

38.7 

39.9 

40.7 

41.1 

13 

7  Orionis 

5  20 

43.1 

41.7 

41.7 

42.1 

42.8 

43.7 

44.6 

45.4 

46.0 

46.4 

14 

e  Orionis 

5  32 

2.4 

1.0 

1.0 

1.3 

2.0 

2.8 

3.7 

4.5 

5.2 

5.5 

15 

a  Orionis 

5  50 

43.1 

41.8 

41.7 

42.0 

42.7 

43.5 

44.4 

45.3 

46.0 

46.4 

16 

a  Argus 

6  22 

9.2 

6.1 

5.5 

5.4 

6.0 

6.9 

8.1 

9.3 

10.2 

10.6 

17 

a  Can.  Maj. 

6  41 

31.6 

30.2 

30.0 

30.1 

30.6 

31.3 

32.2 

33.1 

33.8 

34.3 

18 

e  Can.  Maj. 

6  55 

24.1 

22.6 

22.2 

22.2 

22.6 

23.3 

24.2 

25.2 

26.0 

26.5 

19 

a  Can.  Min. 

7  34 

59.7 

59.0 

58.7 

58.8 

59.1 

59.8 

60.5 

61.5 

62.3 

63.0 

20 

j3  Gemin. 

7  40 

17.1 

16.3 

16.0 

16.0 

16.4 

17.1 

18.0 

19.0 

20.0 

20.8 

21 

e  Argus 

8  20 

51.4 

49.0 

48.0 

47.3 

47.2 

47.8 

48.9 

50.4 

51.8 

52.8 

22 

A  Argus 

9    4 

58.6 

57.9 

57.3 

56.9 

56.8 

57.1 

57.8 

58.9 

60.1 

61.0 

23 

^  Argus 

9  12 

20.6 

18.1 

16.4 

15.1 

14.5 

14.8 

16.0 

17.9 

20.0 

21.7 

24 

a  Hydrse 

9  23 

32.5 

32.6 

32.2 

32.0 

32.0 

32.3 

32.9 

33.7 

34.7 

35.6 

25 

a  Leonis 

10    3 

59.2 

59.7 

59.3 

59.1 

59.0 

59.2 

59.7 

60.5 

61.4 

62.4 

Had  it  been  thus  necessary  to  get  the  deehnation  without 

using  the  chronometer,  we  should  have  proceeded  as  follows  : 

Apparent  solar  time  of  noon  (p.  75) 0'*      0"*  (1) 

Approximate  longitude  =  74°  0'  W.  =  (at  15°  to 

the  hour) 4     56  W.   (2) 

Adding    (1)   and   (2)   (p.  81)   gives  approximate 

Greenwich  apparent  time 4     56  (3) 

Approx,  eq.  of  time,  Dec.  17,  at  4»  SG*"  (p.  76)  +  4  (4) 
Subtracting  ^    (4)    from    (3)    gives    approximate 

G.  M.  T 4     52  (5) 

Deehnation  interpolated  for  G.  M.  T.  in  hne  (5)  is  -  23°   22'  (6) 

^  The  equation  is  additive  to  G.  M.  T.,  according  to  the  note  at 
the  foot  of  p.  76,  and  therefore  to  be  subtracted  from  Greenwich 
apparent  time. 


92 


NAVIGATION   . 


APPARENT   PLACES   OF   STARS,    1917 

From  Nautical  Almanac,  p.  95 
For  the  Upper  Transit  at  Greenwich 


Declination 

Speciaij  Name 

No. 

1— 1 

,_, 

^ 

I— 1 

1—1 

.— I 

^ 

CO 

Mag.i 

0 

+  28 

s 

03 

/ 

38.2 

38.1 

0} 

< 

38.0 

u 

0 

/ 

38.4 

0 

0 

38.5 

1 

1 

38.0 

1 
38.0 

38.5 

38.5 

Alpheratz 

2.2 

2 

+  58 

41.9 

41.8 

41.7 

41.6 

41.5 

42.0 

42.1 

42.2 

42.2 

Caph 

2.4 

3 

-  18 

26.5 

26.5 

26.5 

26.4 

26.3 

26.0 

26.1 

26.2 

26.2 

Deneb  Kaitos 

2.2 

4 

+  59 

48.7 

48.7 

48.6 

48.4 

48.3 

48.6 

48.8 

48.9 

49.0 

Ruchbah 

2.8 

5 

+  88 

52.2 

52.2 

52.1 

52.0 

51.8 

52.0 

52.2 

52.4 

52.5 

Polaris 

2.1 

6 

-57 

39.7 

39.7 

39.6 

39.4 

39.2 

39.0 

39.2 

39.3 

39.4 

Achernar 

0.6 

7 

+  23 

4.5 

4.4 

4.4 

4.3 

4.3 

4.6 

4.7 

4.7 

4.7 

Hamal 

2.2 

8 

-40 

38.3 

38.3 

38.3 

38.2 

38.1 

37.7 

37.8 

38.0 

38.1 

Acamar 

3.0 

9 

+  49 

34.3 

34.3 

34.3 

34.2 

34.1 

34.3 

34.3 

34.4 

34.5 

1.9 

10 

+  16 

20.7 

20.7 

20.7 

20.7 

20.7 

20.8 

20.8 

20.8 

20.8 

Aldebaran 

1.1 

11 

-    8 

17.8 

17.8 

17.9 

17.9 

17.8 

17.5 

17.6 

17.7 

17.7 

Rigel 

0.3 

12 

+  45 

55.0 

55.1 

55.1 

55.1 

55.0 

54.9 

54.9 

55.0 

55.1 

Capella 

0.2 

13 

+    6 

16.6 

16.5 

16.5 

16.5 

16.5 

16.7 

16.7 

16.6 

16.6 

Bellatrix 

1.7 

14 

-    1 

15.2 

15.3 

15.3 

15.3 

15.3 

15.0 

15.1 

15.1 

15.2 

Alnitam 

1.8 

15 

+    7 

23.6 

23.5 

23.5 

23.5 

23.5 

23.7 

23.7 

23.6 

23.6 

Betelgeux 

1.0-1.4 

16 

-  52 

39.0 

39.2 

39.3 

39.3 

39.2 

38.7 

38.7 

38.9 

39.1 

Canopus 

-  0.9 

17 

-  16 

36.1 

36.2 

36.3 

36.3 

36.3 

35.9 

36.0 

36.1 

36.2 

Sirius 

-  1.6 

18 

-  28 

51.5 

51.7 

51.7 

51.8 

51.7 

51.3 

51.4 

51.5 

51.6 

Adhara 

1.6 

19 

+    5 

26.3 

26.2 

26.2 

26.2 

26.2 

26.3 

26.2 

26.2 

26.1 

Procyon 

0.5 

20 

+  28 

13.6 

13.6 

13.6 

13.7 

13.7 

13.5 

13.5 

13.4 

13.4 

Pollux 

1.2 

•21 

-  59 

14.4 

14.6 

14.8 

14.9 

14.9 

14.4 

14.4 

14.5 

14.7 

1.7 

22 

-  43 

5.7 

5.9 

6.1 

6.2 

6.2 

5.8 

5.8 

5.9 

6.0 

2.2 

23 

-  69 

22.4 

22.6 

22.8 

22.9 

23.0 

22.5 

22.4 

22.5 

22.7 

Miaplacidus 

1.8 

24 

-    8 

17.9 

18.1 

18.1 

18.2 

18.2 

18.0 

18.0 

18.1 

18.2 

Alphard 

2.2 

25 

+  12 

22.2 

22.2 

22.2 

22.2 

22.2 

22.2 

22.1 

22.0 

21.9 

Regulus 

1.3 

1  When  the  number  in  this  column  is  very  small,  and  especially  when  it  is  minus, 
the  star  is  very  bright. 

It  is  further  to  be  noted  that  as  we  can  thus  obtain  the 
approximate  G.  M.  T.,  we  really  know  in  advance  the  approx- 
imate moment  when  the  observation  should  be  made.  So 
it  is  unnecessary  to  get  the  sextant  ready  a  long  time  before 
the  observation ;  and  it  is,  in  fact,  better  to  observe  at  the 
proper  predetermined  approximate  moment  rather  than  to 
wait  for  the  maximum  altitude  (p.  86). 

When  the  ship's  position  at  noon  can  be  predicted  with  fair 
approximation,  it  is  thus  possible  to  have  the  declination  and 
other  numbers  for  calculating  the  noon-sight  also  all  ready 


OLDER  NAVIGATION  METHODS  93 

in  advance,  so  that  the  latitude  will  be  immediately  available 
when  the  noon  altitude  has  been  read  from  the  sextant. 

We  shall  now  consider  the  following  example :  Off  St. 
Paul  de  Loando,  West  Africa,  approximate  latitude  8°  55' 
south,  approximate  longitude  12°  55'  east,  both  predicted 
in  advance  by  D.  R.  for  noon  on  Monday,  Dec.  31.  The 
altitude  of  the  sun's  lower  limb  is  to  be  measured.  Index 
correction  is  —  5'.     Height  of  eye,  26  ft. 

To  prepare  for  the  observation,  we  have,  as  before : 

Apparent  solar  time  of  noon 0^     0"»     (1) 

Approximate  D.  R.  longitude  =  12°  55'  east  =  (at  15°  to 

the  hour) 52  E.  (2) 

Subtracting  (2)  from  (1)  gives  approximate  Greenwich 

apparent  time,  Dec.  30 23       8       (3) 

Approximate   equation   of   time,    Dec.   30,  at   23'*  8*" 

(p.  76) _. _ -       3       (4) 

Subtracting  (4)  from  (3),  having  regard  to   —  sign  of 

(4),  gives  approximate  G.  M.  T 23     11       (5) 

The  navigator  will  then  make  the  observation  when  the 
G.  M.  T.  is  23^*  11"*,  as  indicated  by  the  chronometer,  duly 
corrected  for  error  and  rate.  This  would  of  course  also  be 
noon,  or  the  time  when  the  sun  attained  its  maximum  altitude 
for  the  day. 

Now  the  dials  of  chronometers  are  always  divided  into 
12  hours,  like  ordinary  watches,  although  navigators  count 
time  through  24  hours,  as  we  have  seen  (p.  75).  The 
reason  is  that  the  dial  would  be  overloaded  with  numbers 
if  there  were  24  hour  divisions.  Therefore,  when  we  speak 
of  the  chronometer  indicating  23^  11"*,  it  must  be  under- 
stood that  the  actual  chronometer  indication,  or  ''chro- 
nometer face,"  as  it  is  sometimes  called,  would  really  be 
IP  11"* ;  only,  the  navigator  would  call  it  23''  11"*,  astronomic 
time.  In  this  manner  civil  time  still  forces  its  way  into 
navigation,  by  way  of  the  chronometer  face. 

To  make  the  observation  at  the  prearranged  G.  M.  T.  by 
chronometer  it  is  not  desirable  to  carry  that  instrument  out 
into  the  sunlight,  where  the  observer  stands.     It  is  much 


94  NAVIGATION 

better  for  the  navigator  to  use  his  watch,  and  to  calculate  in 
advance  the  ''watch  time"  of  the  observation.  To  do  this, 
it  is  merely  necessary  to  compare  the  watch  with  the  chro- 
nometer, and  thus  ascertain  how  much  the  watch  is  slow  or 
fast  of  the  chronometer.  This  amount  is  called  "chro- 
nometer minus  watch"  (abbreviated  C.  —  W.) ;  and  when  the 
watch  is  fast  of  the  chronometer,  C.  —  W.  is  marked  with  the 
minus  sign. 

To  obtain  the  watch  time  for  the  observation,  we  subtract 
C.  —  W.  from  the  G.  M.  T.  In  the  present  case  we  will 
suppose  the  watch  was  47*"  fast  of  the  chronometer.  Then 
C.  —  W.  =  —  47"*.  To  get  the  watch  time  for  the  observa- 
tion we  must  subtract  —  47*"  from  23''  11*".  Subtracting  a 
minus  number  is  equivalent  to  addition ;  and  so  the  watch 
time  is  23^  11*"  +  47*"  =  23^  58*".  The  observation  would 
be  made  as  nearly  as  possible  2*"  before  noon,  by  the  watch. 

In  this  connection  it  also  becomes  of  interest  to  inquire 
how  the  navigator's  watch  happened  to  be  47*"  fast  of  the 
chronometer.  It  is  customary  aboard  ship  to  set  the  deck 
and  cabin  clocks,  and  all  watches,  to  the  ship's  local  apparent 
time  once  a  day  at  least.     To  do  this,  we  proceed  as  follows  : 

Take  from  chronometer  the  G^  M.  T.,  corrected  for  error  and  rate  (1) 
Apply  to  this  G.  M.  T.  the  eq.  of  time,  giving  Green'h  app.  time  (2) 
Apply  to  (2)  the  approximate  D.  R.  longitude,  adding  it  if  longi- 
tude is  E.,  which  gives  ship's  apparent  time (3) 

And  set  the  watch  to  the  time  (3). 

An  example  of  this  proceeding  can  be  had  from  the  data  on 
p.  93.  Suppose  the  watch  was  to  be  set;  and  the  chro- 
nometer time  was  23^^  0*".  We  should  then  prepare  to  set  the 
watch  in  about  5*",  when  the 

G.  M.  T.  by  chronometer  would  be 23*     5^  (1) 

Chronometer  error  (corrected  for  rate)  say —  2  (2) 

Corrected  G.  M.  T.  by  chronometer,  (1)  +  (2) 23       3  (3) 

Equation  of  time  (p.  93) -  3  (4) 

Greenwich  apparent  time,  (3)  +  (4) 23      0  (5) 

Approximate  longitude  (p.  93) 52  E.  (6) 

Ship's  apparent  time,  (5)  +  (6) 23     52  (7) 


OLDER  NAVIGATION  METHODS  95 

And  the  watch  would  be  set  to  23''  52"*,  when  the  chro- 
nometer face  was  23''  5"" ;  or,  which  is  the  same  thing,  the 
watch  would  be  set  at  8*"  to  12  when  the  chronometer  in- 
dicated 5  minutes  past  11. 

Sometimes  the  navigator  wishes  the  watch  to  be  correct 
by  ship's  apparent  time  at  noon,  but  desires  to  set  it  right 
half  an  hour  sooner,  so  as  to  be  free  at  noon  to  make  an 
observation.  In  that  case  he  calculates  by  D.  R.  what  the 
longitude  will  be  at  noon,  and  proceeds  practically  in  the 
same  way  as  before. 

Resuming  now  the  example  of  p.  93,  we  are  still 
off  St.  Paul  de  Loando,  and  at  2"*  before  noon  by  the 
watch  (p.  94)  the  altitude  of  the  sun's  lower  limb  was 
measured. 

Suppose  it  was  found  to  be 75°    34'  (1) 

The  index  correction  was —  5  (2) 

Adding  (1)  and  (2),  with  regard  to  sign  of  (2),  gives 

corrected  altitude 75     29  (3) 

Correction  from  Table  6 +16  (4) 

Correction  from  Table  7,  for  26  ft.  height  of  eye —  5  (5) 

Adding  (3),  (4),  (5)  gives  corrected  altitude 75     40  (6) 

Formula  (2),  p.  89,  is  the  proper  one,  and  the  inter- 
polated declination,  disregarding  sign,  is 23       8  (7) 

Latitude,  by  formula,  is  (6)  +  (7)  -  90°,  or 8    48  (8) 

The  latitude  of  the  ship  is  therefore  8°  48'  south,  from  the 
above  noon-sight  observation.  The  difference  of  7'  from 
the  approximate  latitude  (p.  93)  might  easily  be  caused  by 
ocean  currents. 

Our  next  example  is  a  star  observation.  Position  of  ship 
by  D.  R.  March  23,  1917,  at  6*  30"»  ship's  time  is :  latitude 
40°  25'  N.,  longitude  46°  52'  W.,  so  that  she  is  near  the  turning 
point  in  the  southern  ''lane  route"  followed  by  steamships 
bound  from  New  York  to  Fast  net  in  summer.  The  upper 
transit  (p.  89)  of  Sirius  was  observed ;  and  the  sextant 
altitude  was  33°  1'.  Index  correction,  —  1' ;  height  of  eye, 
24  ft. 


96  NAVIGATION 

The  calculation  is  as  follows  : 

Observed  altitude  of  Sirius 33°     7'  (1) 

Index  correction —  7  (2) 

Adding  (1)  and  (2),  having  regard  to  minus  sign  of  (2), 

gives  corrected  altitude 33       0  (3) 

Correction  Tables  6  and  7,  combined —  6  (4) 

Adding  (3)  and  (4)  gives  finally  corrected  altitude  ....  32     54  (5) 
Use  formula  (4),  p.  89,  because  latitude  is  +  and  decli- 
nation of  Sirius  — .     We  have 90°  (6) 

Subtract  (5)  from  (6),  giving  (90°  -  altitude) 57       6  (7) 

Declination  of  Sirius  (p.  92),  disregarding  sign,  is.  .  .    16     36  (8) 
Subtract  (8)  from  (7),  giving  (90°—  altitude  —  declina- 
tion), or  the  latitude 40     30  (9) 

Ship's  latitude  at  the  moment  of  observation  was  therefore 
40°  30'  N. 

In  making  such  a  star  observation,  it  is  of  course  possible 
to  follow  the  star  with  the  sextant  until  it  begins  to 
dip  (p.  86)  toward  the  horizon  exactly  as  we  have  ex- 
plained for  the  sun.  But  it  is  preferable  to  prepare  for  the 
observation  in  advance,  and  to  make  it  at  a  definite  prede- 
termined minute  by  the  navigator's  watch.  To  make  such 
preparation,  it  is  necessary  to  use  pages  96  and  97  of  the 
Nautical  Almanac,  parts  of  which  pages  are  reprinted  here 
(pp.  97,  98). 

The  almanac  page  96  gives  for  all  the  bright  stars  the 
G.  M.  T.  of  upper  transit  (p.  89)  at  Greenwich,  for  the  first 
day  of  each  month.  And  it  will  be  noticed  that  the  upper 
transit  is  here  called  ''meridian  transit,"  which  is  practically 
another  name  for  the  same  thing.  Almanac  page  97  (our 
p.  98)  then  gives  a  subtractive  correction,  applicable  to  the 
numbers  on  page  96,  to  make  them  correct  on  days  of  the 
month  other  than  the  1^*. 

Another  small  correction  is  still  required  to  make  the 
numbers  right  in  the  approximate  D.  R.  longitude  of  the  ship, 
instead  of  the  longitude  of  Greenwich,  as  used  on  almanac 
page  96.  This  correction  is  subtractive,  if  the  ship  is  in  west 
longitude,  and  additive,  if  she  is  in  east  longitude ;   and  the 


OLDER  NAVIGATION  METHODS 


97 


MERIDIAN   TRANSIT   OF   STARS,    1917 

From  Nautical  Almanac,  p.  96 

Greenwich  Mean  Time  of  Transit  at  Greenwich 


Constella- 

y-^ 

.-< 

-H 

- 

y-^ 

1— 1 

1-H 

rH 

- 

tion 

Mag. 

z 

n 

« 

« 

< 

a. 

H 

> 

Name 

< 

H 

fe 

^ 

< 

i 

o 

^ 

Q 

h  m 

h  m 

h  m 

h  m' 

h 

m 

h 

m 

h  m 

h  m 

h  m 

a  Androm. 

2.2 

5  21 

3  19 

1  29 

23  23 

21 

25 

13 

22 

11  24 

9  22 

7  24 

/3  Cassiop. 

2.4 

5  22 

3  20 

1  30 

23  24 

21 

26 

13 

22 

11  24 

9  22 

7  24 

^  Ceti 

2.2 

5  56 

3  54 

2  4 

f  0   2) 
123  68i 

22 

0 

13 

57 

11  59 

9  57 

7  59 

8  Cassiop. 

2.8 

6  37 

4  35 

2  45 

0  43 

22 

41 

14 

38 

12  40 

10  38 

8  40 

a  Urs.  Min. 

2.1 

6  47 

4  45 

2  54 

0  52 

22 

50 

14 

49 

12  51 

10  49 

8  51 

a  Eridani 

0.6 

6  51 

4  49 

2  59 

0  57 

22 

55 

14 

52 

12  54 

10  52 

8  54 

a  Arietis 

2.2 

7  19 

5  17 

3  27 

1  25 

23 

23 

15 

20 

13  22 

11  20 

9  22 

6  Eridani 

3.0 

8  12 

6  10 

4  20 

2  18 

0 

20 

16 

12 

14  14 

12  12 

10  14 

o  Persei 

1.9 

8  35 

6  33 

4  43 

2  41 

0 

43 

16 

35 

14  38 

12  36 

10  38 

a  Tauri 

1.1 

9  47 

7  46 

5  55 

3  54 

1 

56 

17 

48 

15  50 

13  48 

11  50 

/3  Orionis 

0.3 

10  27 

8  25 

6  35 

4  33 

2 

35 

18 

27 

16  29 

14  28 

12  30 

a  Aurigae 

0.2 

10  27 

8  25 

6  35 

4  33 

2 

35 

18 

27 

16  29 

14  28 

12  30 

V  Orionis 

1.7 

10  37 

8  35 

6  45 

4  43 

2 

45 

18 

37 

16  39 

14  38 

12  40 

e  Orionis 

1.8 

10  48 

8  46 

6  56 

4  54 

2 

56 

18 

49 

16  51 

14  49 

12  51 

a  Orionis 

1.0-1.4 

11  7 

9  5 

7  15 

5  13 

3 

15 

19 

7 

17  9 

15  7 

13  9 

a  Argus 

-0.9 

11  38 

9  36 

7  46 

5  44 

3 

46 

19 

39 

17  41 

15  39 

13  41 

a  Can.  Maj. 

-1.6 

11  57 

9  55 

8  5 

6  3 

4 

5 

19 

58 

18  0 

15  58 

14  0 

e  Can.  Maj. 

1.6 

12  11 

10  9 

8  19 

6  17 

4 

19 

20 

12 

18  14 

16  12 

14  14 

a  Can.  Min. 

0.5 

12  51 

10  49 

8  59 

6  57 

4 

59 

20 

51 

18  53 

16  52 

14  54 

^  Gemin. 

1.2 

12  56 

10  54 

9  4 

7  2 

5 

4 

20 

57 

18  59 

16  57 

14  59 

€  Argus 

1.7 

13  36 

11  34 

9  44 

7  42 

5 

44 

21 

37 

19  39 

17  37 

15  39 

A  Argus 

2.2 

14  20 

12  19 

10  28 

8  27 

6 

28 

22 

21 

20  23 

18  21 

16  23 

/3  Argus 

1.8 

14  28 

12  26 

10  36 

8  34 

6 

36 

22 

28 

20  30 

18  28 

16  31 

a  Hydrce 

2.2 

14  39 

12  37 

10  47 

8  45 

6 

47 

22 

40 

20  42 

18  40 

16  42 

a  Leonis 

1.3 

15  19 

13  17 

11  27 

9  25 

7 

27 

23 

20 

21  22 

19  20 

17  22 

amount  of  it  is  10*  for  every  15°  in  the  ship's  longitude. 
After  it  has  been  apphed,  the  result  will  be  the  ship's  mean 
solar  time  of  the  star's  upper  transit. 

As  an  example,  let  us  take  the  preparation  for  the  fore- 
going observation  of  Sirius,  or  a  Can.  Maj.     We  have  : 

G.  M.  T.  of  upper  transit,  March   1,   from  almanac 

page  96  above 8^     5"*  (1) 

Correction    for  23d    day  of    month,    from    almanac 

page  97  (our  p.  98) -  1     27     (2) 

Correcting  (1)  with  (2),  having  regard  to  —  sign  of  (2)  6  38  (3) 
Further  correction  for  longitude  46°  52'  W.,  at  10«  per 

15°  of  longitude,  approximately 1      (4) 

Subtracting  (4)  from  (3)  gives  ship's  mean  solar  time 

of  the  observation 6    37     C5) 

H 


98 


NAVIGATION 


MERIDIAN    TRANSIT    OF    STARS,    1917 
From  Nautical  Almanac,  p.  97 

Corrections  to  be  Applied  to  the  Mean  Time  of  Transit  on 
THE  First  Day  of  the  Month,  to  Find  the  Mean  Time  of 
Transit  on  any  Other  Day  of  the  Month 


Day  of 
Month 

Correction 

Day  of 
Month 

Correction 

Day  of 
Month 

Correction 

h        m 

h        m 

h        m 

1 

-0       0 

11 

-0     39 

21 

-  1     19 

2 

0       4 

12 

0    43 

22 

1     23 

3 

0       8 

13 

0    47 

23 

1     27 

4 

0     12 

14 

0     51 

24 

1     30 

5 

0     16 

15 

0     55 

25 

1     34 

6 

-0     20 

16 

-0     59 

26 

-  1     38 

7 

0     24 

17 

1       3 

27 

1     42 

8 

0     28 

18 

1       7 

28 

1     46 

9 

0     31 

19 

1     11 

29 

1     50 

10 

0     35 

20 

1     15 

30 

1     54 

11 

-0     39 

21 

-  1     19 

31 

-  1     58 

Note.  If  the  quantity  taken  from  this  Table  is  greater  than  the 
mean  time  of  transit  on  the  first  of  the  month,  increase  that  time 
by  23'*  56"»  and  then  apply  the  correction  taken  from  this  Table. 


The  actual  observation  was  made  at  6*  30*",  ship's  time, 
as  indicated  by  the  navigator's  watch.  The  difference  of 
7*"  between  6*  30*",  and  6^^  37*"  in  line  (5)  above,  is  due  to  the 
equation  of  time  (p.  77),  which  is  7*"  on  March  23.  This 
7*",  if  applied  (with  its  proper  sign  from  the  almanac)  to 
line  (5)  above,  will  give  the  ship's  apparent  time;  and  we 
have  seen  that  watches  and  clocks  on  board  are  usually 
kept  set  to  apparent  and  not  mean  ship's  time  (p.  94). 

To  complete  this  part  of  our  subject,  we  have  still  to  con- 
sider a  few  additional  points  of  interest.  For  instance,  a 
star  chosen  for  observation  may  be  one  of  the  planets : 
Mars,  Jupiter,  or  Saturn.  These  look  Hke  very  bright  stars 
in  the  sextant  telescope ;  and  calculations  depending  on 
them  are  similar  to  those  described  for  stars.  The  planetary 
declinations  and  the  G.  M.  T.'s  of  their  upper  transits  are 
given  in  the  almanac,  but  not  on  the  pages  reprinted  here. 


OLDER  NAVIGATION  METHODS  99 

The  moon  is  now  so  rarely  observed  that  we  have  not  given 
examples  of  lunar  observations. 

Sometimes  an  ''ex-meridian"  observation  of  the  sun  or 
a  star  is  made  at  a  time  very  near  the  upper  transit,  on  a 
day  when  the  actual  transit  observation  could  not  be  secured 
because  of  clouds.  There  are  special  tables  ^  for  calculating 
observations  of  this  kind;  but  we  have  not  included  them 
here  because  all  such  observations  can  be  satisfactorily 
treated  by  a  new  general  method  to  be  explained  later 
(p.  108). 

Having  now  fully  treated  the  older  standard  method  of 
determining  the  ship's  latitude,  let  us  next  consider  the  older 
way  of  obtaining  the  longitude.  This  cannot  be  done  when 
the  sun  (or  a  star)  is  near  its  maximum  altitude,  as  already 
explained  (p.  88).  The  most  favorable  opportunity  occurs 
when  the  observed  object  bears  (p.  44)  east  or  west ;  but 
it  is  not  always  possible  to  get  the  observation  on  such  a 
bearing.  In  that  case,  the  longitude  observation,  often 
called  a  ''time-sight,"  must  be  taken  when  the  sun  is  near 
the  desired  bearing,  but  always  avoiding,  if  possible,  observa- 
tions at  very  low  altitudes.  And  if  a  very  low  altitude  has 
been  observed  in  an  emergency,  it  can  sometimes  be  checked 
by  a  later  observation  at  a  better  altitude. 

The  principle  on  which  the  time-sight  depends  is  simple. 
Calculations  based  on  the  measured  altitude  make  known 
the  ship's  mean  time  at  the  moment  of  observation.  At 
the  same  moment  the  chronometer  face  (p.  93),  duly  cor- 
rected for  error  and  rate,  tells  us  the  G.  M.  T.  The 
difference  between  the  two  times  then  gives  us  the  longitude 
(see  p.  82). 

The  calculations  for  this  problem  are  made  by  means  of 
Table  4  (trigonometric  logarithms)  and  Table  10  ("haver- 
sines").  These  haversines  (abbreviated  ha  v.)  are  really 
additional  trigonometric  logarithms ;  and  Table  10  gives 
in  every  case  not  only  the  haversine  itself,  which  is  really 
1  Tables  26  and  27  of  Bowditch's  "Navigator,"  for  instance. 


100  NAVIGATION 

a  logarithm,  but  also,  in  the  adjoining  heavy  type  col- 
umns, the  number  (abbreviated  No.)  of  which  the  haver- 
sine  is  the  log.  This  additional  heavy  type  number  is  not 
given  throughout  the  entire  table,  but  only  when  necessary 
for  working  Sumner  line  calculations  (see  Chapter  IX, 
p.  108).     It  is  not  needed  in  working  time-sights. 

The  argument  (p.  10)  of  the  haversine  table  is  a  double 
argument,  not  to  be  confounded  with  the  pairs  of  arguments 
already  explained  (p.  11).  In  the  haversine  table,  the  argu- 
ment is  generally  given  in  degrees  and  minutes,  as  well  as 
(for  convenience)  in  hours  and  minutes  of  time,  allowing 
the  usual  15°  to  each  hour,  etc. 

We  shall  now  solve  our  time-sight  problem  for  the  sun ; 
and  in  doing  so  shall  make  use  of  two  angles  not  hitherto 
employed:  the  ''polar  distance''  (abbreviated  p),  and  the 
''half  sum"  (abbreviated  s).  We  shall  also,  for  brevity, 
indicate  the  ship's  apparent  solar  time  by  T.  Then  we 
have  the  following  formulas  : 

If  lat.  and  dec.  are  both  +  or  both  —  .  .p  =  90°  —  dec.  (1) 

If  lat.  and  dec.  are  one  +  and  one  —  .  .  .p  =  90°  +  dec.  (2) 

In  every  case s  =  ^  (alt.  +  lat.  +  p)  (3) 

If  time-sight  was  made  before  noon,  ship's  time, 

hav.  (24^  —  7")  =  see  lat.  +  esc  p  +  cos  s  +  sin  (s  —  alt.)  (4) 
If  time-sight  was  made  after  noon,  ship's  time, 

hav.  T  =  sec  lat.  +  esc  p  +  cos  s  +  sin  (s  —  alt.)  (5) 

In  using  these  formulas,  we  have  to  choose  between  (1) 
and  (2),  and  also  between  (4)  and  (5).  Formula  (3)  is 
always  used.  No  attention  need  be  given  to  the  signs 
of  the  declination  or  latitude  except  in  choosing  between 
formulas  (1)  and  (2)  for  calculating  p;  and  in  choosing 
between  (4)  and  (5),  we  have  merely  to  note  whether  the 
time-sight  was  taken  in  the  forenoon  or  afternoon  by  ship's 
time. 

We  also  desire  to  emphasize  especially  that  these  formulas 
presuppose  the  latitude  to  be  known.  This  is  merely 
another  application  of  the  principle  (p.  88)  that  both  lati- 


OLDER  NAVIGATION  METHODS  101 

tude  and  longitude  cannot  be  determined  from  a  single 
observation.  It  follows  that  in  using  this  method  we  must 
first  determine  the  latitude  by  a  noon-sight  before  we  can 
calculate  the  time-sight  for  longitude.  If  the  time-sight 
was  taken  in  the  afternoon,  the  noon-sight  will  naturally 
have  preceded  it,  and  the  ship's  latitude  at  noon  will  be 
known.  This  noon  latitude  must  then  be  carried  forward 
to  the  moment  of  the  afternoon  time-sight  by  D.  R.  methods 
(p.  7) ;  and  the  latitude  thus  obtained  must  be  used  for 
calculating  the  time-sight. 

But  if  the  time-sight  was  a  forenoon  observation,  it  cannot 
be  properly  calculated  until  noon,  when  the  latitude  will 
be  determined.  After  that,  the  latitude  can  be  carried 
backwards  by  D.  R.  to  the  moment  of  the  forenoon  time- 
sight,  and  the  latter  can  be  calculated. 

But  if  the  navigator,  because  of  emergency,  needs  his 
longitude  at  once,  after  taking  the  forenoon  time-sight,  he 
must  obtain  the  latitude  by  a  D.  R.  calculation  based  on  the 
last  good  noon-sight.  Most  navigators  calculate  morning 
time-sights  in  this  way,  and  then  repeat  the  calculation 
after  the  new  noon-sight  has  been  obtained.  The  latter 
calculation  will  be  preferable  to  the  former,  because  the 
further  the  latitude  is  carried  along  by  D.  R.,  the  less  accurate 
will  it  be.  And  any  error  in  the  latitude  used  in  the  calcula- 
tion will  impress  a  consequent  error  on  the  calculated  longi- 
tude. 

We  shall  now  work  some  time-sight  examples.  On  board 
ship,  at  sea,  Dec.  18,  1917,  in  the  afternoon,  D.  R.  latitude 
42°  20'  N.,  D.  R.  longitude  35°  16'  W.,  the  altitude  of  sun's 
lower  limb  was  observed  to  be  14°  19'.  The  time  was  taken 
with  the  navigator's  watch,  and  was  2^  29^  58*.  A  com- 
parison of  the  watch  and  ship's  chronometer  gave  C.  —  W.  = 
2h  27"!  8*.  The  chronometer  correction  was  2*"  8*  slow  of 
G.  M.  T.  The  index  correction  of  the  sextant  was  -(-  4' ; 
height  of  eye,  24  ft.     Calculate  the  ship's  longitude. 

We  have  first  to  find,  for  the  moment  of  the  observation. 


102  NAVIGATION 

values  of  the  declination  and  equation  of  time.     To  do  this, 
we  have : 

Watch  time  of  observation 2»   29"»  58*  (1) 

C.  -  W 2     27       8  (2) 

Adding  (1)   and  (2)  gives  chronometer  time  of 

observation 4     57       6  (3) 

Chronometer  correction,  slow 2       8  (4) 

Adding  (3)  and  (4)  gives  G.  M.  T.  of  observation  4     59     14  (5) 

For  the  G.  M.  T,  (5)  we  interpolate  the  declina- 
tion (p.  76),  finding -  23°    24'  (6) 

and  for  the  same  G.  M.  T.  we  interpolate  the 

equation  of  time +  3"»   21«  (7) 

Now,  adding  (5)  and  (7)  gives  Greenwich  ap- 
parent time  of  observation 5^    2"*  35*  (8) 

Next  we  inspect  the  formulas  (p.  100),  choosing  (2)  be- 
cause latitude  is  +  and  declination  — ,  and  (5)  because  the 
sight  was  an  afternoon  one. 

We  now  have,  from  line  (6),  declination  (disregard- 
ing sign) 23°  24'  (9) 

to  which,  by  formula  (2),  we  add 90      0  (10) 

giving  p 113     24  (11) 

The  observed  altitude  was 14     19  (12) 

Index  correction '. +4  (13) 

Adding  (12)  and  (13)  gives  corrected  altitude 14     23  (14) 

Correction,  Table  6 +12  (15) 

Correction,  Table  7 -  5  (16) 

Adding  (14),  (15),  (16)  gives  finally  corrected  altitude  14     30  (17) 

The  latitude  by  D.  R.  is 42     20  (18) 

Adding  (11),  (17),  (18)  gives 170     14  (19) 

Halving  (19)  gives  (by  formula  (3),  p.  100)  s 85       7  (20) 

Subtracting  (17)  from  (20)  gives  (s  -  alt.) 70    37  (21) 

Next  we  apply  formula  (5),  p.  100.     We  have: 

sec  lat.  (18)  from  Table  4,  page  238 0.13121  (22) 

CSC  p  (11)  from  Table  4,  page  219 0.03727  (23) 

cos  s  (20)  from  Table  4,  page  200 8.93007  (24) 

sin  (s  -  alt.)  (21)  from  Table  4,  page  215 9.97466  (25) 

sum  (22)  to  (25)  =  hav.  T,  by  formula  (5) 9.07321  ^  (26) 

1  This  sum  has  been  diminished  by  10  arbitrarily  (see  p.  25), 
which  must  always  be  done  when  the  sum  of  logs  is  larger  than  10. 


OLDER  NAVIGATION  METHODS  103 

7,1  corresponding  to  (26)  from.  Table  10,  page  260,  is  2^*  40"»  59«  (27) 
Greenwich    apparent    time    (8)    by    watch    and 

chronometer  is 5       2     35    (28) 

Subtract   (27)  from  (28),  giving  time  difference 

between  ship  and  Greenwich 2     21     36    (29) 

Turning  (29)  into  degrees  with  Table  9,  page  249, 

gives 35°  24'    W.   (30) 

and  (30)  is  the  ship's  longitude  from  this  time-sight. 

Upon  comparing  the  D.  R.  longitude  (35°  16'  W.)  with  the 
result  of  the  time-sight  (35°  24'  W.),  we  find  that  the  ship 
is  8'  west  of  her  D.  R.  position.  This  means,  of  course,  that 
there  has  been  a  westerly  ''set"  of  current  in  the  interval 
between  the  last  accurate  determination  of  longitude  and 
the  present  one.  It  would  be  proper  for  the  navigator  to 
calculate  from  this  the  amount  of  westerly  drift  per  hour, 
and  to  allow  for  it  in  carrying  forward  his  longitude  by  D.  R. 
from  the  present  time-sight.  It  is  also  clear  that  the 
northerly  or  southerly  set  of  the  current  can  be  similarly 
measured  and  allowed  for  by  comparing  the  D.  R.  latitude 
with  the  latitude  from  a  noon-sight  (cf.  p.  95).  It  is  the 
general  custom  of  navigators  to  ascribe  such  differences  to 
ocean  currents,  never  to  uncertainty  in  the  astronomic  results. 
Dead  reckoning  is  never  allowed  any  weight  as  against  a 
sextant  observation. 

The  reader  will  have  noticed  that  the  foregoing  calculation 
has  been  made  in  great  detail,  so  that  a  beginner  may  have 
no  difficulty  in  understanding  it.  But  a  practiced  navigator 
would  of  course  work  the  calculation  in  a  much  more  con- 
densed form,  in  such  a  way  as  to  bring  the  logarithms  next 
to  the  numbers  to  which  they  belong.  We  shall  therefore 
now  repeat  the  same  example  in  such  a  condensed  form : 

1  If  the  observation  had  been  made  before  noon,  we  should  have 
used  formula  (4)  and  should  here  have  obtained  24^  —  T,  instead 
of  T.  This  24^  -  T  would  then  be  subtracted  from  24^^,  to  get 
T,  before  continuing  the  calculation.  Thus  the  form  of  calculation 
would  contain  another  line  between  (27)  and  (28),  in  the  case  of 
a  forenoon  observation. 


104 


NAVIGATION 


TIME-SIGHT,  CONDENSED  FORM.    SUN 


Watch  time  :  2^   29»»  58*  (1) 

C.  -  W. :  2     27  8  (2) 

Chr.  time :  4     57  6  (3) 

Chr.  corr'n :  +     2  8  (4) 

G.  M.  T. :  18tb  4     59  14  (5) 

Eq.  of  time  :  +3  21  (7) 

G.  app.  time :  5       2  35  (8) 


Obs'd  alt. : 
Index : 
Table  6 : 
Table  7 : 
Corr'd  alt. 


14°  19'  (12) 
+  4  (13) 
+  12  (15) 
-    5    (16) 

14  30    (17) 


Decl.  18th,  4» :    23°  23'.7 
H.  D. :  0.1 

Decl.  4'*  59'" :      23     24         (6) 
p:  113     24       (11) 


Eq.  time,  18^^,  4^* :  +  3"»  22».3 
H.  D. :  1.2 

Eq.  time,  4^*  59^ :    +3      21.1 


(7) 


Corr'd  alt. 

:          14°  30'  (17) 

Lat.,  D.  R. 

42    20    (18)     sec  lat. : 

0.13121   (22) 

p: 

113    24    (11)     CSC  v- 

0.03727  (23) 

sum  of  3 : 

2)170    14    (19) 

5 : 

85      7    (20)     COS  s : 

8.93007  (24) 

s  —  alt. : 

70    37    (21)     sin  (s-alt.) 

:  9.97466  (25) 

sum  of  4 : 

9.07321  (26)  = 

hav.  T 

(or  24" 

-  r)i 

T  =  ship's  app.  time : 

2*   40"»  59* 

(27) 

By  chron.. 

Greenwich  app.  time : 

5       2     35 

(8) 

Longitude : 

2'»   21'"  36* 

(29) 

or: 

35°   24'    W. 

(30) 

When  the  object  observed  is  a  star  or  planet,  the  choice 
between  formulas  (4)  and  (5),  p.  100,  is  not  quite  the  same 
as  in  the  case  of  a  solar  time-sight.  We  must  use  (4)  if  there 
is  any  east  in  the  star's  bearing  at  the  moment  of  observation  ; 
and  (5),  if  there  is  west  in  the  bearing.  The  more  nearly  the 
star  bears  due  east  or  west,  the  more  accurate  will  be  the 
resulting  longitude.  The  use  of  formulas  (1),  (2),  and  (3) 
is  the  same  as  for  the  sun ;  but  T,  in  the  case  of  a  star,  is  no 
longer  the  ship's  apparent  solar  time.     Instead,  it  is  called 


^  See  p.  103,  footnote. 


OLDER   NAVIGATION  METHODS  105 

the  star's  ''hour-angle."  To  get  the  longitude,  we  must 
first  (p.  85)  calculate  the  Greenwich  sidereal  time  corre- 
sponding to  the  G.  M.  T.  of  the  observation,  as  taken  from 
the  chronometer,  duly  corrected  for  error  and  rate ;  and 
then  use  the  following  formulas : 

(6)  Greenwich  sid.  time^—  right-ascension  of  star  =  Greenwich 
hour-angle. 

,-     I  West  long.  =  Greenwich  hour-angle  —  T, 
\  East  long.  =  T  —  Greenwich  hour-angle. 

As  an  example  of  a  star  observation  we  shall  take  the 
following : 

At  sea,  just  before  sunrise,  Dec.  17,  1917,  off  Cape  Agulhas, 
latitude  by  D.  R.  35°  20'  S.,  longitude  by  D.  R.  20°  41'  E., 
the  altitude  of  Sirius  was  measured,  and  found  to  be  40°  3'. 
The  star  bore  west,  and  the  height  of  eye  was  22  ft.  Index 
correction  was  +  5'.  Time  by  watch,  IQ^  29*"  48%  or  4^  29"» 
48*  A.M.,  civil  time,  Dec.  18;  C.  -  W.,  -  I''  23^  50%-  chro- 
nometer fast  of  G.  M.  T.  2"^  28^ 

The  calculation  would  proceed  thus : 

Watch  time  of  observation 16*   29'«  48»     (1) 

C.  -  W -  1    23     50      (2) 

Adding  (1)  and  (2),  having  regard  to  —  sign  of  (2), 

gives  chronometer  time  of  observation 15       5     58      (3) 

Chronometer  correction,  fast —  2     28      (4) 

Adding  (3)  and  (4),  having  regard  to  —  sign  of  (4), 

gives  G.  M.  T.  of  observation 15       3     30      (5) 

Right  ascension  mean  sun,  Greenwich  mean  noon, 

Dec.  17  (p.  83) 17     42     10      (6) 

Correction  for  "  time  past  noon  "  (see  p.  84) ....  2     28      (7) 

Adding  (6)  and  (7)  gives  right  ascension  of  mean 

sun 17     44      38     (8) 

Adding  (5)  and  (8)   (see  p.  85)  gives  Greenwich 

sidereal  time  of  the  observation 8  ^   48       8      (9) 

Right  ascension  of  Sirius,  Dec.  17,  is  (p.  91).  ...     6     41     34    (10) 

Subtracting  (10)  from  (9)  gives  Greenwich  hour- 
angle  (formula  (6),  above) 2       6     34    (11) 

1  24''  may  always  be  added  or  dropped  here,  if  necessary. 


106  NAVIGATION 

Next  we  calculate  T  by  formula  (5),  p.  100.     We  have: 

Declination  of  Sirius,  Dec.  17  (p.  92)    -  16°  36'  (12) 

By  formula  (1),  p.  100,  subtract  (12)  from  90°, 

without  attention  to  sign  of  (12), giving  p.  .  73    24  (13) 

The  observed  altitude  was 40      3  (14) 

The  index  correction  was +5  (15) 

Table  6  correction —  1  (16) 

Table  7  correction —  5  (17) 

Adding   (14),   (15),  (16),  (17),  having  regard  to 

signs,  gives  corrected  altitude 40      2  (18) 

The  latitude  by  D.  R.  was 35    20  (19) 

Adding  (13),  (18),  and  (19)  gives 148    46  (20) 

Halving  (20)  gives  s 74    23  (21) 

Subtracting  (18)  from  (21)  gives  (s  -  altitude)  .  .  34    21  (22) 


Now  applying  formula  (5),  page  100,  we  have  : 

sec  latitude  (19)  from  Table  4,  page  231   0.08842  (23) 

CSC  V  (13)  from  Table  4,  page  212 0.01849  (24) 

cos  s  (21)  from  Table  4,  page  211 9.43008  (25) 

sin  (s  -  altitude)  (22)  from  Table  4,  page  230 9.75147  (26) 

Summing  (23)  to  (26)  gives  hav.  T,  by  form.  (5)  .  .9.28846  ^  (27) 
7^2 corresponding  to  (27),  from  Tab.  10,  p.  263  is .  .  3*  29«  14*  (28) 
Difference  between  (28)  and  (11)  is  the  longi- 
tude by  formula  (7),  page  105 1    22     40   E.  (29) 

Turning  (29)  into   degrees  with  Table  9,  page 

249,  gives 20°  40'  E.  (30) 

The  D.  R.  longitude,  20°  41'  E.,  was  therefore  within  1'  of 
the  longitude  from  this  time-sight,  and  this  shows  that  the 
ship  has  not  been  affected  by  ocean  currents  since  the  last 
observation.  It  is  also  interesting  to  note  how  near  sunrise 
the  observation  was  made.  The  twilight  must  have  been 
quite  strong,  and  the  star  therefore  dim.  But  star  observa- 
tions can  be  made  best  in  twilight  because  the  horizon  Hne 
can  then  be  seen  distinctly. 

^  This  sum  has  also  been  diminished  by  10  (see  footnote,  p.  102). 
2  Might  be  24*  -  T,  if  the  star  bore  E.  instead  of  W.  (see  footnote, 
p.  103). 


OLDER  NAVIGATION  METHODS 


107 


The  foregoing  example  can  of  course  also  be  arranged  in 
condensed  form,  as  follows : 


TIME-SIGHT, 

CONDENSED 

FORM. 

STAR 

Watch  time : 

16^^  29« 

48» 

(1) 

Obs'd  alt. : 

40°     3' 

(14) 

C.  -  W. : 

-1 

23 

50 

(2) 

Index : 

+  5 

(15) 

Chr.  time : 

15 

5 

58 

(3) 

Table  6 : 

-1 

(16) 

Chr.  corr'n : 

- 

-  2 

28 

(4) 

Table  7 : 

-5 

(17) 

G.  M.  T. : 

15 

3 

30 

(5) 

Corr'd  alt. 

:  40       2 

(18) 

R.  A.  mean  sun : 

17 

42 

10 

(6) 

Lat.  D.  R. 

:35     20 

(19) 

Corr'n,  past  noon : 

2 

28 

(7) 

p: 

73     24 

(13) 

Greenw'h  sid.  time : 

8 

48 

8 

(9) 

sum:        2)148     46 

(20) 

R.  A.  of  Sirius : 

6 

41 

34 

(10) 

s : 

74     23 

(21) 

Green\vdch  hour-ang. 

:  2 

6 

34 

(11) 

(s-alt.): 

34     21 

(22) 

T.,  from  (27) : 

3 

29 

14 

(28) 

Long. : 

1 

22 

40 

E.  (29) 

or: 

20° 

40' 

E.  (30) 

R.  A.  of  Sirius : 

6'^ 

41m    34, 

(10) 

Dec.  of  Sirius : 

— 

16°    36' 

(12) 

p: 

73     24 

(13) 

sec  lat. : 

0.08842 

(23) 

CSC.  p : 

0.01849 

(24) 

cos  s : 

9.43008 

(25) 

sin  (s  -  alt.) : 

9.75147 

(26) 

sum  of  4 : 

9.28846  (27)  =  hav.  T 

(or  24*  - 

T)i 

Having  now  fully  explained  both  the  noon-sight  and  the 
time-sight,  we  shall  close  this  chapter  with  a  strong  recom- 
mendation to  young  navigators  to  familiarize  themselves  with 
the  observation  of  stars.  These  always  furnish  a  valuable 
check  on  sun  observations  :  and  at  times  of  danger  may  save 
the  ship  when  clouds  have  obscured  the  sun  for  days,  and 
clearing  occurs  after  sunset.  It  is  easy  to  learn  to  know  the 
principal  stars  from  Jacoby's  ''Astronomy,"  Chapter  III, 
"How  to  Know  the  Stars." 

1  See  footnote,  p.  103. 


CHAPTER   IX 
NEWER  NAVIGATION   METHODS 

The  reader  may  have  noticed  in  Chapter  VIII  that  there 
is  a  very  definite  difference  between  the  determination  of 
latitude  by  a  noon-sight  and  longitude  by  a  time-sight :  for 
the  latitude  is  obtained  without  previous  knowledge  of  the 
longitude ;  but  to  get  the  longitude,  a  previous  knowledge 
of  the  latitude  is  essential.  This  is,  of  course,  a  decided 
disadvantage  in  determining  longitude,  nor  is  there  any 
practicable  direct  way  to  get  the  longitude  without  first 
knowing  the  latitude. 

We  have  also  seen  (p.  101)  that  any  existing  uncertainty 
in  our  knowledge  of  the  latitude  will  produce  an  error  in  the 
longitude  computed  from  a  time-sight.  In  situations  of 
danger  it  is  important  to  ascertain  how  great  this  longitude 
error  may  be.  Suppose,  for  instance,  we  have  calculated 
a  time-sight  with  a  D.  R.  latitude  that  we  suspect  may  be 
as  much  as  10'  too  small ;  and  we  wish  to  know  how  much 
our  computed  longitude  may  have  been  thereby  put  wrong. 
The  obvious  way  to  find  out  is  to  recompute  the  longitude 
with  an  assumed  latitude  10'  larger  than  the  D.  R.  latitude. 
The  resulting  longitude  will  then  show  the  extreme  range 
of  error  that  must  have  been  produced  if  the  D.  R.  latitude 
was  10'  too  small. 

A  third  calculation,  with  an  assumed  latitude  10'  smaller 
than  the  D.  R.  latitude,  will  similarly  exhibit  the  extreme 
possible  range  of  longitude  error  in  the  other  direction. 
Thus  these  two  extra  calculations  will  show  the  limits  of 
longitude  error  that  might  be  caused  by  a  range  of  20'  in 
the  possible  error  of  the  D.  R.  latitude. 

108 


NEWER  NAVIGATION   METHODS  109 

This  rather  obvious  procedure  was  probably  used  long 
ago  by  more  than  one  intelligent  navigator ;  but  it  was  first 
published  in  1837  by  Thomas  H.  Sumner,  an  American 
merchant  captain.  He  used  the  method  in  dramatic  cir- 
cumstances of  great  danger ;  and  he  brought  his  ship  safely 
into  port.  According  to  his  own  account,  he  made  three 
calculations  of  the  longitude,  using  three  assumed  latitudes 
differing  by  10',  and  he  of  course  obtained  three  different 
longitudes.  He  then  marked  or  plotted  (p.  55)  on  his  chart 
the  point  indicated  by  the  first  assumed  latitude  and  its 
computed  longitude.  At  this  point  the  ship  must  have  been 
located,  if  the  first  assumed  latitude  had  been  correct.  The 
other  two  latitudes,  with  their  computed  longitudes,  indicated 
two  more  points  on  the  chart ;  and  at  one  of  these  points  the 
ship  must  have  been,  if  either  of  these  additional  latitudes 
was  correct. 

Sumner  found  that  the  three  points  on  the  chart  lay  in  a 
straight  line;  and  it  became  at  once  evident  that  whatever 
latitude  he  might  assume  (within  reason)  he  would  always 
get  a  point  on  the  same  straight  line,  after  computing  the 
longitude.  .  In  other  words,  although  he  did  not  know  his 
latitude  accurately,  and  so  could  not  compute  his  longitude 
accurately,  yet  he  had  found  a  straight  line  on  the  chart 
upon  which  his  ship  was  surely  situated. 

Such  a  line  can  always  be  found  in  the  way  Sumner  found 
it,  or  in  some  preferable  modern  way;  and  such  a  line  we 
shall  call  a  ''Sumner  line,"  though  some  writers  on  naviga- 
tion prefer  to  call  it  a  ''line  of  position." 

On  the  occasion  of  laying  down  his  line,  Sumner  found  that 
it  passed  directly  through  Small's  Light,  near  the  Irish  coast  ; 
and  as  the  line  bore  E.N.E.  on  his  chart,  he  simply  put 
the  ship  on  that  course,  and  in  less  than  an  hour  he  "made" 
Small's  Light,  actually  bearing  E.N.E.  J  E.,  and,  as  he  says, 
"close  aboard."  He  had  had  no  observations  after  passing 
longitude  21°  W.,  until  the  morning  of  Dec.  17,  when  these 
historic  events  occurred.     He  was  off  a  rocky  lee  shore,  in 


no  NAVIGATION 

the  midst  of  a  winter  gale,  after  crossing  the  Atlantic ;  only 
a  seaman  can  understand  the  relief  he  must  have  felt  when 
that  light  suddenly  appeared  off  the  bow. 

We  have  given  this  account  of  Sumner's  experience  to 
impress  on  the  young  navigator  that  he  must  positively 
familiarize  himself  with  the  Sumner  method  of  navigation. 
Should  we  be  so  fortunate  as  to  have  any  experienced  navi- 
gator among  our  readers,  we  ask  him  to  try  the  Sumner 
method  once  more,  in  the  manner  explained  below,  even  if 
he  may  have  found  it  troublesome  in  the  past  on  account  of 
certain  difficulties  in  its  application.  For  the  Sumner 
method  is  the  best  method  of  navigation  on  all  oceans  and 
at  all  times  :  even  when  a  noon-sight  is  available  for  latitude, 
it  is  better  to  treat  it  as  a  Sumner  observation,  and  work 
out  the  Sumner  line. 

The  principal  objection  urged  against  it  by  certain  prac- 
tical navigators  arises  from  the  small  scale  of  existing  ocean 
track  charts,  on  which  a  distance  of  10'  is  represented  by 
about  ^  inch.  A  line  like  Sumner's,  20'  long,  would  have 
only  a  length  of  J  inch  on  the  chart ;  and  such  a  little  line 
would  not  be  long  enough  to  show  accurately  the  direction 
in  which  it  pointed.  When  near  a  coast,  as  in  Sumner's 
case,  this  difficulty  disappears,  because  navigators  always 
have  (or  always  should  have  and  use)  the  large  scale  charts 
that  can  be  obtained  for  coastwise  waters. 

But  it  is  inconvenient  for  navigators  to  begin  using  a 
method  off  the  coast,  on  the  last  day  of  a  voyage,  different 
from  the  form  employed  for  many  days  at  sea.  Therefore, 
some  authorities  recommend  the  construction  of  a  special 
large  scale  chart,  with  its  latitude  and  longitude  lines,  each 
time  an  observation  is  made  throughout  the  voyage,  so  that 
the  Sumner  line  can  always  be  drawn  on  a  sufficiently  large 
scale.  It  is  no  wonder  that  navigators  have  not  generally 
adopted  this  somewhat  laborious  proceeding ;  and  in  the 
method  given  below  we  shall  utilize  the  Sumner  idea  without 
requiring  any  lines  to  be  drawn  on  charts. 


NEWER  NAVIGATION  METHODS  111 

Another  objection  to  Sumner  navigation  is  that  it  requires 
too  much  calculation ;  three  longitude  calculations  for  one 
observation,  as  Sumner  practiced  it.  This  objection  is  also 
quite  removed  now  by  the  use  of  suitable  tables  such  as  we 
give  in  the  present  volume. 

But  before  proceeding  to  explain  these  tables,  we  must 
outline  briefly  the  real  principle  on  which  rests  the  com- 
plete utilization  of  the  Sumner  method  on  the  open  sea. 
There  the  navigator  wants  to  know  the  ship's  position  in 
both  latitude  and  longitude ;  and  will  not  be  satisfied  with 
a  mere  line,  with  the  ship  ''somewhere  on  the  line."  Along 
the  coast  such  a  line  might  help  him  to  find  Small's  Light ; 
but  he  is  not  looking  for  coast  lights  at  sea. 

And  the  Sumner  method  takes  care  of  this  matter  in  the 
simplest  possible  way.  We  have  seen  (p.  88)  that  two 
different  observations  are  always  necessary  by  any  method 
to  get  both  latitude  and  longitude.  But  two  such  observa- 
tions by  the  Sumner  method  give  two  different  lines  on  the 
chart :  and  as  the  ship  must  be  located  on  both  lines,  her 
actual  position  must  be  at  their  point  of  intersection.  We 
shall  show  how  the  required  latitude  and  longitude  of  the 
ship  at  the  point  of  intersection  can  be  found  by  a  simple 
calculation,  without  the  drawing  of  any  lines  on  the  chart. 

Coming  now  to  the  modern  method  of  calculating  a  Sum- 
ner line,  we  must  first  state  a  general  fundamental  principle 
that  may  be  easily  verified  by  geometrical  considerations. 
The  true  bearing  (p.  44)  of  a  Sumner  line  on  a  chart  is 
always  90°  greater  than  the  true  bearing  or  azimuth  (p.  44) 
of  the  sun  (or  star)  at  the  moment  of  observation.  Or,  in 
other  words,  the  Sumner  line  bears  at  right  angles  to  the 
sun  at  the  time  of  observation. 

We  shall  show  how  the  bearing  or  azimuth  of  the  sun  can 
always  be  found  from  suitable  ''azimuth  tables";  but  the 
Sumner  line  is  not  completely  known  from  its  bearing  alone. 
To  locate  it  properly  it  is  necessary  to  know  in  addition  the 
latitude  and  longitude  of  some  point  on  the  line,  which  we 


112  NAVIGATION 

will  call  a  ''Sumner  point."  Then,  knowing  such  a  point  oi 
the  line,  and  the  bearing  of  the  line,  we  may  say  we  know  the 
line  completely,  and,  if  necessary,  could  draw  it  on  a  chart. 

Now  to  find  the  required  Sumner  point.  We  always  have 
the  D.  R.  position  of  the  ship  at  the  moment  of  observation ; 
which  we  will  call  the  ''D.  R.  point."  It  is  easy  to  find 
out  if  the  D.  R.  point  is  also  a  Sumner  point.  It  is  merely 
necessary  to  calculate  what  the  sun's  altitude  would  be  for 
a  ship  at  the  D.  R.  point,  and  then  compare  this  calculated 
altitude  with  the  one  actually  observed.  If  the  D.  R.  point 
was  really  a  Sumner  point  (which  will  rarely  happen),  the 
two  altitudes  will  agree ;  if  not,  the  amount  of  disagreement 
will  show  how  far  the  D.  R.  point  is  distant  from  the  nearest 
Sumner  point. ^ 

The  first  step,  then,  in  Sumner  navigation,  is  the  calcula- 
tion of  the  altitude,  supposing  the  ship  to  be  at  the  D.  R. 
point  at  the  moment  of  observation.  To  do  this  for  a  sun 
observation,  we  first  calculate  the  Greenwich  apparent  time 
(abbreviated  G.  A.  T.)  of  the  observation,  just  as  was  done 
in  the  case  of  a  time-sight  on  p.  102.  To  this  G.  A.  T.  we 
then  add  the  ship's  D.  R.  loi;igitude,  if  east,  or  Subtract  it,  if 
west,  to  get  T  (p.  100),  the  ship's  apparent  time  of  the  ob- 
servation. We  then  use  the  formulas  on  p.  113,  in  which 
X  and  Z  are  ''auxiliary  angles"  required  in  the  calculations, 
but  not  otherwise  of  special  interest.  These  formulas  are 
called  the  "  cosine-haver  sine  "  formulas. 

There  are  several  other  sets  of  formulas  with  which  the 
same  problem  can  be  solved.  One  set,  called  the  "  haversine  " 
formulas,  involves  the  use  of  haversines  only;  another, 
called  the  "  sine-cosine "  formulas,  solves  the  problem  with 
sines  and  cosines.  But  neither  is  preferable  to  the  following 
cosine-haversine  set. 

1  This  method  is  often  called  the  Marcq  Saint  Hilaire  method ; 
but  it  should  probably  be  credited  to  Lord  Kelvin,  who  published 
"  Tables  for  Facilitating  Sumner's  Method  at  Sea  "  in  1876.  These 
tables  follow  the  method  described  above. 


NEWER   NAVIGATION  METHODS  113 

If  observation  was  made  before  noon,  ship's  time, 

hav.  X  =  cos  lat.  +  cos  dec.  +  hav.  (24''  -  T),  (1) 

If  observation  was  made  after  noon,  ship's  time, 

hav.  X  =  cos  lat.  +  cos  dec.  +  hav.  T,  (2) 

lat.  —  dec.  =  diff.^  of  lat.  and  dec,  if  both  are  +  or  both  — ,  (3) 

lat.  —  dec.  =  sum^  of  lat.  and  dec.  if  one  is  +  and  one  — ,  (4) 

No.  hav.  Z  =  No.  hav.  (lat.  -  dec.)  +  No.  hav.  X,  (5) 

Alt.  =  90°  -  Z.  (6) 

Now  we  can  compare  the  altitude  computed  by  formula 
(6)  with  the  observed  altitude,  fully  corrected  for  index 
error,  etc.  The  difference  between  the  two  altitudes  in 
minutes  will  be  the  distance  in  miles  of  the  nearest  Sumner 
point  from  the  D.  R.  point,  for  the  minute  and  nautical 
mile  here  correspond,  as  they  do  in  the  case  of  differences  of 
latitude  (p.  15).  The  bearing  of  the  Sumner  point  from  the 
D.  R.  point  will  be  the  same  as  the  sun's  azimuth  if  the  ob- 
served altitude  is  greater  than  the  computed  altitude  :  but  if 
the  observed  altitude  is  less  than  the  computed,  the  bearing  of 
the  Sumner  point  will  be  180°  greater  than  the  sun's  azimuth. 

The  bearing  and  distance  of  the  Sumner  point  from  the 
D.  R.  point  once  known,  it  is  easy,  by  means  of  the  traverse 
table  (p.  10),  to  obtain  the  latitude  and  longitude  of  the 
Sumner  point  from  the  known  latitude  and  longitude  of 
the  D.  R.  point ;  or,  which  is  the  same  thing,  from  the  ship's 
D.  R.  latitude  and  longitude. 

Before  giving  examples  of  these  calculations,  it  remains 
to  show  how  the  sun's  bearing  or  azimuth  can  be  taken  from 
Table  11  (p.  284),  called  the  azimuth  table.  The  pair  of 
arguments  (p.  11)  for  entering  this  table  are:  first,  in  the 
left-hand  column,  the  declination,  which  is  here  used  without 
regard  to  its  sign ;  and  second,  in  the  four  topmost  hori- 

1  In  using  formulas  (3)  and  (4),  pay  no  attention  to  +  or  — 
signs  after  the  right  formula  is  once  chosen.  The  difference  between 
latitude  and  declination  is  always  taken  by  subtracting  the  smaller 
from  the  larger ;  and  the  sum  by  adding  them,  without  regarding 
their  +  or  —  signs.     Cf.  also  p.  89. 


114  NAVIGATION 

zontal  lines,  T  (p.  100),  the  ship's  apparent  time  at  the 
moment  of  observation. 

Having  found  this  pair  of  arguments,  we  look  in  the 
column  under  T,  and  in  the  horizontal  line  opposite  the 
declination.  There  we  find  an  '4ndex  number."  Next  we 
look  up  the  altitude,  as  computed  by  formula  (6),  page  113, 
in  the  right-hand  column  of  the  azimuth  table,  and  follow 
along  the  horizontal  line  belonging  to  that  altitude,  until 
we  reach  a  number  equal  (or  nearly  equal)  to  the  index 
number.  Then  we  go  down  the  column  containing  this 
second  appearance  of  the  index  number,  and  find  the  azi- 
muth at  the  bottom  of  the  page.  The  table  gives  approxi- 
mate azimuths  only,  but  the  approximation  is  sufficient  for 
our  present  purpose. 

The  azimuths  at  the  bottom  of  the  page  appear  in  four 
horizontal  lines,  of  which  the  upper  two  belong  to  forenoon 
observations,  and  the  lower  two  to  afternoon  observations. 
All  azimuths  are  counted  from  the  north,  through  east, 
south,  and  west,  from  0°  to  360°,  like  compass  courses  in 
United  States  Navy  practice  (p.  41).  It  is  important  for 
the  navigator  to  record,  at  the  time  of  observation,  the  word 
''forenoon"  or  '^ afternoon,"  and  also  the  sun's  roughly 
approximate  bearing,  to  aid  in  choosing  which  of  the  azi- 
muths at  the  bottom  of  the  tabular  page  is  the  right  one. 
The  record  showing  whether  the  observation  was  made  in 
the  forenoon  or  afternoon  limits  the  choice  to  two  of  the  lines 
of  azimuths;  and  if  there  is  any  doubt  remaining  between 
these  two,  the  following  rules  may  clear  it  up. 

When  latitude  is  +  and  declination  — ,  azimuth  is  between 
90°  and  270°  ; 

When  latitude  is  +  and  declination  +,  if  declination  is 
greater  than  latitude,  azimuth  is  not  between  90°  and  270° ; 

When  latitude  is  —  and  declination  — ,  if  declination  is 
greater  than  latitude,  azimuth  is  between  90°  and  270° ; 

When  latitude  is  —  and  decHnation  -j-,  azimuth  is  not 
between  90°  and  270°. 


NEWER  NAVIGATION  METHODS  115 

In  other  cases,  and  especially  when  latitude  and  declina- 
tion are  nearly  equal,  the  foregoing  rules  are  insufficient,  and 
we  must  consult  Table  12  (p.  290),  the  ''auxiliary  azimuth 
table."  This  table  has  latitude  and  declination  for  its  pair 
of  arguments,  the  former  in  the  left-hand  vertical  column, 
the  latter  in  the  topmost  horizontal  line :  and  in  using  the 
table  it  is  not  necessary  to  pay  attention  to  the  +  and  — 
signs  of  latitude  and  declination.  Start  with  the  latitude, 
and  follow  its  horizontal  line  to  the  right  until  you  reach  the 
column  having  the  declination  at  its  head.  There  you  will 
find  an  ''auxiliary  angle,"  which  must  be  compared  with 
the  altitude  computed  by  formula  (6),  page  113.     Then : 

If  the  computed  altitude  is  greater  than  the  auxiliary 
angle,  and  if  latitude  is  -\-,  azimuth  is  between  90°  and  270° ; 

If  the  computed  altitude  is  less  than  the  auxiliary  angle, 
and  if  latitude  is  — ,  azimuth  is  between  90°  and  270° ; 

If  the  computed  altitude  is  less  than  the  auxiliary  angle, 
and  if  latitude  is  +,  azimuth  is  not  between  90°  and  270° ; 

If  the  computed  altitude  is  greater  than  the  auxiliary 
angle,  and  if  latitude  is  — ,  azimuth  is  not  between  90°  and 
270°. 

It  will  rarely  happen  that  any  of  the  foregoing  rules  will 
be  needed,  if  the  navigator  will  make  a  careful  observation 
of  the  sun's  azimuth  with  the  azimuth  circle  or  pelorus 
(p.  44),  as  soon  as  possible  after  the  sextant  altitude  has 
been  observed.  The  ship's  course  should  also  be  specially 
recorded  when  this  observation  is  made.  This  proceeding 
is  not  merely  a  convenience  to  avoid  consulting  the  fore- 
going rules  in  using  the  azimuth  table :  it  is  really  essential 
to  safe  navigation,  for  a  comparison  of  the  observed  azi- 
muth with  that  derived  from  the  table  will  make  the  com- 
pass error  (p.  43)  known.  The  variation  is  known  from  the 
chart ;  so  that  if  we  observe  the  compass  error,  we  can  allow 
for  the  variation,  and  get  the  deviation.*  This  can  then  be 
compared  with  the  deviation  table  (p.  48),  to  see  if  there  has 
been  any  change  in  the  compass  since  leaving  port.     It  is 


116  NAVIGATION 

a  great  advantage  of  the  Sumner  method  that  the  sun's 
azimuth  comes  out  as  a  sort  of  by-product,  so  that  the  com- 
pass can  be  verified  without  any  additional  special  calcu- 
lations. 

We  shall  now  illustrate  all  the  above  considerations  by 
means  of  examples ;  beginning  with  the  observation  already 
treated  as  a  time-sight  (p.  101).  That  observation  we  shall 
now  work  by  the  Sumner  method.  From  page  101  we  take 
the  following : 

Date  of  observation,  Dec.  18,  1917,  in  the  afternoon;  D.  R. 
latitude,  42°  20' N. ;  D.  R.  longitude,  35°  16'  W. ;  altitude  observed, 
14°  19' ;  time  by  watch,  2^  29"»  58* ;  C.  -  W.,  2''  27'"  8* ;  chronometer 
correction,  2"*  8*  slow  of  G.  M.  T. ;  index  correction,  +  4' ;  height  of 
eye,  24  ft. 

From  the  preparatory  part  of  the  calculation  (p.  102), 
we  also  copy  the  following  additional  numbers : 

Declination,  line  (6),  page  102 -23°  24'     (1) 

Greenwich  apparent  time  (G.  A.  T.)  of  observation, 

Une  (8),  page  102 5*  2«  35-     (2) 

We  have  next  to  calculate,  by  the  formulas  on  page  113,  the 
altitude  corresponding  to  the  D.  R.  point,  for  which  the 
latitude  and  longitude  are  given  above.  The  longitude  is 
35°  16'  W.,  or,  at  15°  to  the  hour  (Table  9,  p.  249) : 

D.  R.  longitude  is 2*  21'"  4«W.  (3) 

Subtracting  (3)  from  (2),  according  to  page  112, 

gives  ship's  apparent  time  of  observation,  T.  .    2    41     31       (4) 

We  are  now  prepared  to  apply  formulas  (1)  to  (6), 
page  113.  We  choose  formula  (2)  for  an  afternoon  obser- 
vation ^ ;   and  write  : 

1  For  a  forenoon  observation  we  should  choose  formula  (1),  and 
should  therefore  need  to  know  24*  — T  instead  of  T.  This  would 
make  necessary  another  line  in  the  form  of  calculation,  and  it  would 
follow  line  (4).  This  new  line  might  be  numbered  (4') ;  and  in  it 
would  be  written  24*  —  T,  obtained  by  subtracting  T  (line  4)  from 
24*. 


NEWER  NAVIGATION  METHODS  117 

Cos  lat.,  42°  20'  N.  by  D.  R.  (see  Table  4,  p.  238) ....  9.86879  (5) 

Cos  dec,  23"  24',  line  (1)  (see  Table  4,  p.  219) 9.96273  (6) 

Hav.  T,  2^  41"*  31*,  line  (4)  (see  Table  10,  p.  260) ....  9.07596  (7) 
Adding  (5)  to  (7)  gives  hav.  X  (dropping  20,  p.  25) .  .   8.90748  (8) 

Now  we  choose  formula  (4),  because  latitude  and  declina- 
tion are  +  and  —  ; 

The  latitude  is,  by  D.  R 42"  20'     (9) 

Adding   (1)   and   (9)  according  to  formula  (4)  gives 

(lat.  -  dec.) 65°  44'  (10) 

Now  we  have.  Table  10,  page  266,  No.  hav.  of  (10) . .  0.29451   (11) 

No.  hav.  X,i  line  (8) 0.08082  (12) 

Adding  (11)  and  (12),  according  to  formula  (5),  page 

113,  gives  No.  hav.  Z 0.37533  (13) 

And  Z,  corresponding  to  (13)  is  found  from  Table  10, 

page  268 75°  34'  (14) 

Then,  by  formula  (6)  computed  altitude  =90°  —  Z  (14), 

or 14°  26'   (15) 

This  computed  altitude  (15)  must  now  be  compared  with 

the  observed  altitude,  fully  corrected.     We  find  : 

Obs'd  alt.,  fully  corrected,  line  (17),  page  102,  is 14°  30'   (16) 

Difference  between  (15)  and  (16),  in  minutes,  is  the 
distance  of  Sumner  point  from  D.  R.  point  in 
miles  (p.  113).     It  is 4  miles  (17) 

Next  we  must  find  the  sun's  azimuth  from  Table  11,  page 
286.  The  top  argument  for  entering  the  table  is  T,  line 
(4),  and  it  must  be  found  in  the  ''afternoon"  lines.  The 
argument  for  the  left-hand  column  is  the  declination,  line  (1). 
Under  T,  and  opposite  declination,  we  find  the  tabular  index 
number  5872.^  Then  we  find  the  computed  altitude,  line 
(15),  in  the  right-hand  column  of  Table  11,  page  286,  and 

1  This  No.  hav.  X  comes  from  Table  10,  page  258,  without  looking 
up  the  angle  X  at  all.  We  simply  find  hav.  X  in  the  table,  and  take 
the  No.  hav.  X  out  of  the  adjoining  heavy  type  column.  No  inter- 
polations are  needed,  the  nearest  tabular  numbers  being  sufficiently 
accurate. 

2  The  index  numbers  and  the  azimuth  need  not  be  very  accurate  : 
it  is  sufficient  to  use  the  nearest  tabular  arguments,  so  that  inter- 
polation is  not  essential. 


118  NAVIGATION 

follow  its  horizontal  line  till  we  again  come  upon  the  index 
number  5872.  It  lies  about  halfway  between  5703  and 
5973.  Going  down  the  two  columns  containing  these  index 
numbers,  we  find  in  the  afternoon  azimuth  lines  two  values 
of  the  azimuth,  217°  and  323°.  The  choice  between  these 
two  numbers  would  be  very  easy,  if  the  observer's  record 
contained  even  a  rough  estimate  of  the  sun's  bearing  at  the 
time  of  observation.  We  have  purposely  not  made  this  avail- 
able, so  as  to  show  how  to  consult  the  directions  on  page 
114,  and  there  we  find  that  when  the  latitude  is  +  and  the 
declination  — ,  the  azimuth  is  between  90°  and  270°.  So 
we  finally  choose  217°  for  the  sun's  azimuth. 

Since  the  observed  altitude  (16)  is  greater  than  the  com- 
puted altitude  (15),  the  bearing  of  the  Sumner  point  from 
the  D.  R.  point,  according  to  page  113,  is  the  same  as  the  sun's 
azimuth,  or  217°.  And  as  we  now  know  the  bearing  and 
distance  of  the  Sumner  point  from  the  D.  R.  point,  we  can 
find  its  latitude  and  longitude  by  a  simple  application  of  the 
traverse  table  (p.  154). 

We  have  merely  to  consider  the  bearing  and  distance  to 
be  a  course  angle  and  distance,  and  imagine  a  ship  to  have 
sailed  from  the  one  point  to  the  other.  In  the  present  case, 
the  distance  is  4  miles  (line  17),  the  course  217°  :  and  Table  1 
(p.  164)  gives  the  corresponding  latitude  3'. 2,  departure  2.4. 
The  longitude  difference  is  obtained  from  the  departure  by 
Table  2  (p.  174)  and  is,  for  latitude  42°,  about  3'.2.  Drop- 
ping odd  fractions,  the  latitude  difference  and  longitude  differ- 
ence both  come  out  3'.  The  Sumner  point  is  therefore  3'  dis- 
tant from  the  D.  R.  point  in  both  latitude  and  longitude. 
And  since  the  bearing  217°  indicates  on  the  compass  card 
that  the  Sumner  point  is  south  and  west  of  the  D.  R.  point, 
it  follows  that : 

Lat.  of  Sumner  point  =  D.  R.  lat.  —  3'  = 

42°  20'  N.  (line  9)  -  3' 42°  17'  N.  (18) 

Long,  of  Sumner  point  =  D.  R.  long.  +3' 35   19  W.  (19) 

Azimuth  of  Sumner  line  (p.  Ill) 307°  (20) 


NEWER  NAVIGATION  METHODS 


119 


It  is  important  for  the  reader  to  understand  that  the  fore- 
going calculation  is  given  in  extended  detail  so  as  to  make 
it  easy  for  the  beginner  to  follow.  In  condensed  form, 
we  should  have  the  following  arrangement  of  the  calculation, 
corresponding  to  the  condensed  time-sight  form  (p.  104). 
Part  of  the  work  here  repeated  from  page  104  has  no  attached 
reference  numbers  in  parentheses  :  the  new  part  of  the  work 
has  references  to  the  detailed  calculation  just  given. 


SUMNER  LINE,  CONDENSED  FORM.    SUN 


Obs'd  alt.:   14° 

19' 

Decl.  4'' :         23° 

23'.7  S., 

Index :              + 

4 

H.  D.: 

0.1 

Table  6 :          +12 

Decl.  4^^  59"» :  23° 

24'  S. 

Table  7  : 

5 

Eq.  time,  4* : 

+  3'^  22«.3 

Corr'dalt. :    14° 

30' 

H.  D.: 

Eq.  time,  4^  59'» 

1.2 
:    +3     21.1 

Watch  time : 

2'k29"» 

58« 

C.  -  W. : 

2   27 

8 

Chr.  time : 

4   57 

6 

Chr.  corr'n : 

+ 

2 

8 

G.  M.  T.  18th : 

4   59 

14 

Eq.  of  time : 

+ 

3 

21 

G.  app.  time : 

5     2 

35 

D.  R.  long. : 

2    21 

4W.    (3) 

Ship's  app.  time, 

T: 

2    41 

31 

(4) 

hav.  T  (or  24^^  - 

TY:   9.07596 

D.  R.  lat. : 

42°  20' 

N. 

(9) 

cos  lat. : 

9.86879 

Dec: 

23    24 

S. 

(1) 

cos  dec. : 
sum  =  hav.  X : 
No.  hav.  X : 
No.  hav.  (lat. 

9.96273 

8.90748 
0.08082  (12) 

Lat.  —  Dec. : 

65   44 

(10) 

-  dec.)  : 

0.29451  (11) 

Z: 

75  34 

(14) 

No.  hav.  Z 

0.37533  (13) 

Comp'd  alt. : 

14   26 

(15) 

Obs'd  alt. : 

14   30 

(16) 

Diff.: 

4 

(17) 

Index  No. : 

5872 

Azimuth : 

217° 

Lat.  difif. : 

3'.2 

Dep.: 

Long.  diff. : 

2.4 
3'.2 

D.  R.  lat. : 

42°  20' 

N. 

(9) 

D.  R.  long. : 

35°  16'  W.     (3) 

Sumner  pt.  lat. : 

42   17 

N. 

(18) 

Sumner  pt.  long. : 

35    19  W.  (19) 

1  See  footnote,  p.  116. 


120  NAVIGATION 

When  the  object  observed  is  a  star  (cf.  p.  104)  or  planet, 
the  choice  between  formulas  (1)  and  (2),  page  113,  is  not  quite 
the  same  as  in  the  case  of  a  solar  observation.  We  must 
use  formula  (1)  if  the  star  was  on  the  east  side  of  the  sky- 
when  observed,  which  might  be  called  a  ''forenoon"  observa- 
tion of  the  star ;  and  we  must  use  (2)  if  the  star  was  on  the 
west  side  of  the  sky,  giving  an  ''afternoon"  star  observa- 
tion. The  use  of  the  remaining  formulas  (3)  to  (6)  is  the 
same  as  for  the  sun  ;  but  T  is  now  no  longer  the  ship's  appar- 
ent time.  Instead,  it  is  the  star's  hour-angle  (p.  104) ; 
to  find  it  for  use  in  formulas  (1)  and  (2),  and  in  Table  11, 
we  must  first  calculate  (p.  85)  the  Greenwich  sidereal 
time  corresponding  to  the  G.  M.  T.  of  the  observation,  as 
taken  from  the  chronometer,  duly  corrected  for  error  and 
rate ;   and  then  use  the  following  formulas  : 

(7)  Greenwich  hour-angle  =  Greenwich  sidereal  time  —  right  ascen- 
sion of  star, 
T  =  Greenwich  hour-angle  +  D.  R.  longitude,  if  east, 


(8) 

I  r  =  Greenwich  hour-angle  —  D.  R.  longitude,  if  west. 

As  an  application  of  the  Sumner  method  to  a  star  observa- 
tion, let  us  take  the  observation  of  Sirius,  Dec.  17,  1917, 
off  Cape  Agulhas,  already  treated  as  a  time-sight  (p.  105). 

From  the  preliminary  calculations  there  given,  we  have : 

Greenwich  hour-angle,  line  (11),  page  105 2^     6*"  34*       (1) 

D.   R.  longitude   (p.   105)   is  20°  41'  E.,  or  by 

Table  9  (p.  249) 1     22     44  E.   (2) 

By  formula    (8)    above,   we   add    (1)   and    (2), 

giving  T 3     29     18        (3) 

The  star  bore  west  ^  (p.  105)  so  we  choose  formula  (2) 

(p.  113),  and  write : 

cos    lat.    (p.    106,    line    19),   35°  20'    S.    by    D.   R. 

(see  Table  4,  p.  231) 9.91158  (4) 

cos  dec.  (p.  106,  line  12),   -  16°  36'  (Tab.  4,  p.  212)  9.98151  (5) 

hav.  r,  3*  29*"  18*  (line  3,  above)  (see  Table  10,  p.  263)  9.28872  (6) 

Adding(4)to(6)gives,byformula(2),pagell3,hav.Z,  9.181812  (7) 

^  See  p.  116,  footnote. 

2  Sum  diminished  by  20  (see  footnote,  p.  102). 


NEWER  NAVIGATION  METHODS  121 

Next  we  choose  formula  (3),  page  113,  since  latitude  and 
declination  are  both  — .     We  have  : 

By  formula  (3),  lat.  -  dec.  =  35°  20'  -  16°  36'  =         18°  44'      (8) 

We  now  use  formula  (5),  page  113.     We  have: 

No.  hav.  18°  44'  (8)  (see  Table  10,  p.  254) 0.02649     (9) 

No.  hav.  XI  (7)  (see  Table  10,  p.  261) 0.15194  (10) 

Adding  (9)  and  (10)  gives  No.  hav.  Z 0.17843  (11) 

And  Z,  corresponding  to  (11)  is  found  from 

Table  10,  page  262 49°  59'    (12) 

Then,  by  formula  (6),  page  113, 

computed  alt.  =  90°  -  Z  (12),  or 40°    1'     (13) 

This  computed  altitude  (13)  must  be  compared 
with  the  observed  altitude,  fully  corrected. 

This  was  (p.  106,  line  18) 40°    2'     (14) 

Difference  between  (13)  and  (14),  in  minutes,  or  dis- 
tance of  Sumner  point  from  D.  R.  point  in  miles 
(p.  113)   1  mile       (15) 

Next  we  find  the  star's  azimuth  from  Table  11,  page  287. 

The  top  argument  for  entering  the  table  is  T,  line  (3), 
and  it  must  be  found  in  the  "afternoon"  lines,  since  the  star 
bore  W.  The  argument  for  the  left-hand  column  is  the 
declination,  line  (5).  Under  T  (p.  287),  and  opposite 
declination,  we  find  (approximately)  the  tabular  index  num- 
ber 7550.  Then  we  find  the  computed  altitude,  40°  (13), 
in  the  right-hand  column  of  the  table  (p.  289),  and  follow 
along  its  horizontal  line  until  we  again  reach  the  index 
number  7550.  The  nearest  to  7550  is  7544;  and  under 
this  number,  at  the  foot  of  the  column,  we  find  the  two 
"afternoon"  azimuths  260°  and  280°. 

These  two  numbers  are  so  nearly  equal  that  there  is  un- 
certainty in  choosing  between  them.  Had  the  observer 
taken  the  star's  bearing  by  compass  at  the  time  of  observa- 
tion (p.  115),  the  uncertainty  would  be  removed.  But 
in  the  absence  of  this  information,  we  must  have  recourse 
to  Table  12  (p.  290),  the  auxiliary  azimuth  table.  Enter- 
ing this  table  with  the  pair  of  arguments  of  the  present 

^  No.  hav.  here  obtained  from  hav.  without  finding  the  angle  X 
(p.  117,  footnote). 


122  NAVIGATION 

problem:  viz.  latitude  35°,  declination  17°,  we  find  the 
auxiliary  angle  31°.  The  computed  altitude  (13)  being 
40°,  is  greater  than  the  auxiliary  angle,  and  the  latitude  is  — . 
Therefore,  by  the  instructions  (p.  115),  the  azimuth  is 
not  between  90°  and  270°.  We  therefore  choose  280°  as 
our  final  azimuth,  since  260°,  the  other  possible  value,  is  in 
the  prohibited  area  between  90°  and  270°. 

The  computed  altitude  (13)  being  less  than  the  observed 
altitude,  this  observation  places  the  Sumner  point  1  mile 
(15)  from  the  D.  R.  point,  and  bearing  from  it  280°,  the  same 
as  the  star's  azimuth  (p.  113).  The  traverse  table  (p.  156) 
gives,  for  distance  1  and  course  280°,  latitude  0.2,  departure 
1.0.  The  longitude  difference,  by  Table  2  (p.  172),  is  r.2, 
for  the  departure  1 .0.  Therefore,  since  azimuth  280°  indicates 
on  the  compass  card  that  the  Sumner  point  is  W.  and  N. 
of  the  D.  R.  point,  we  have : 

lat.  of  Sumner  point  =  -  35°  20'  (4)  +  0'.2  =  -  35°  20'  (16) 
long,  of  Sumner  point  =       20°  41'  E.  (2)  -  1'.2  =  20°  40'  E.    (17) 

The  bearing  of  the  Sumner  Jine  will  be  90°  greater  than 
the  star's  azimuth  (p.  Ill) ;  so  we  have  : 

Bearing  of  Sumner  line  =  280°  +  90°  =  370° ;  or, 

dropping  360°  =    10°  (18) 

The  foregoing  calculation  of  the  Sumner  point  from  a 
star  observation  can  of  course  also  be  put  in  condensed  form. 
In  doing  so,  we  have  repeated  certain  numbers  from  page  107 
without  references  in  parentheses.  But  numbers  taken 
from  the  extended  calculation  just  given  have  their  reference 
numbers  attached. 

This  condensed  form,  like  the  others  previously  given,  is 
the  form  of  calculation  which  would  be  used  in  actual 
navigation.  It  is  most  important,  in  the  interest  of  numeri- 
cal accurac}^,  to  make  all  calculations  upon  forms ;  and  no 
numbers  should  be  written  on  the  forms  without  having  an 
adjoining  statement  as  to  the  meaning  of  the  numbers. 


NEWER  NAVIGATION  METHODS 


123 


SUMNER   LINE,    CONDENSED   FORM.     STAR 


Watch  time  :                    16*   29*"  48* 

C.  -  W. :                      -  1     23     50 

Chr.  time :                       15       5     58 

Chr.  corr'n  :                          -  2     28 

Obs'd  alt. : 

40°  3' 

G.  M.  T. :                       15       3     30 

Index : 

+  5 

R.  A.  mean  sun :             17     42     10 

Table  6 : 

-  1 

Corr'n,  past  noon :                   2     28 

Table  7 : 

-  5 

Greenw'h  sid.  time :        8     48       8 

Corr'd  alt. : 

40    2 

R.  A.  of  Sirius :                6     41     34 

Greenw'h  hour-angle :     2       6     34 

D.  R.  long. :                      1     22     44  E. 

(2) 

T:                                     3     29     18 

(3) 

T  or  (24»  -  T)  1 :  3'»  29'"   18» 

(3)  hav 

. :  9.28872 

(6) 

Dec. :                   -  16°  36' 

cos 

:     9.98151 

(5) 

D.  R.  lat. :          -  35     20 

cos 

:     9.91158 

(4) 

Sum  of  3  =  hav.  X : 

9.18181 

(7) 

No.  hav.  X : 

0.15194 

(10) 

Lat.  -  Dec. :           18°  44'    (8) ; 

;  No.  hav. :  0.02649 

(9) 

Sum  of  2  =  No.  hav.  Z : 

0.17843 

(11) 

Z: 

49°  59' 

(12) 

Computed  alt.  =  90°  -  Z : 

40     1 

(13) 

Obs'd  alt.,  corr'd: 

40     2 

(14) 

Diff. : 

1 

(15) 

Index  No. :  7550 

Azimuth :     280° 

Lat.  diff. :  0'.2         Dep. :  1.0 

Long. 

diff. :  1'.2 

Sumner  pt.  lat. :  -  35°  20'  (16) 

;   long. 

:  20°  40'  E. 

.   (17) 

Bearing  of  Sumner  line  :   10°  (18) 


We  have  now,  in  the  foregoing  examples,  illustrated  the 
manner  of  determining  a  Sumner  line  completely  by  ascer- 
taining the  latitude  and  longitude  of  one  point  on  the  line 
(the  Sumner  point),  and  the  bearing  of  the  line  itself  at  that 
point.  It  may  be  desired  to  draw  the  line  on  the  chart, 
which  will  always  interest  the  navigator  if  he  is  near  the 
coast  and  has  a  large-scale  chart.  To  draw  it,  we  merely 
locate  the  Sumner  point  on  the  chart  by  its  latitude  and  longi- 

1  See  footnote,  p.  116. 


124  NAVIGATION 

tude,  and  then  draw  the  line  through  the  point  so  that  it 
will  make  with  the  meridian  an  angle  equal  to  the  bearing 
which  has  been  computed  for  the  line.  The  Sumner  line 
should  be  extended  in  both  directions  from  the  Sumner 
point,  for  any  convenient  distance,  in  such  a  way  that  the 
point  will  be  near  the  middle  of  the  line. 

We  can  now  gain  a  better  understanding  as  to  Sumner 
navigation  by  comparing  the  results  obtained  in  one  of  the 
foregoing  examples  with  the  corresponding  calculation  of 
the  same  example  as  a  time-sight.     Thus  from  the  same  ob 
servation  (pp.   104,  119) 

As  A  Time-Sight  As  a  Sumner  Observation 

From  D.  R.  latitude  42°  20'  N. ;  From  D.  R.  latitude 42°  20'  N. ; 
D.  R.  longitude  35°  16'  W.,  we  D.  R.  longitude  35°  16'  W.,  we 
found  the  ship's  longitude  to  be  found  the  Sumner  point  to  be 
35°  24'  W.  in  latitude  42°  17' ;  longitude  35° 

19'  W. ;  and  azimuth  of  Sumner 

line,  307°. 

Starting  with  the  same  observed  altitude,  and  the  same 
D.  R.  position  of  the  ship,  we  get  quite  different  results  by 
the  two  methods  of  calculation.  The  time-sight  gives  us 
nothing  but  a  longitude ;  and  it  will  be  the  correct  ship's 
longitude  only  if  the  D.  R.  latitude  was  also  correct  (p.  101). 
Therefore  the  time-sight  calculation  leaves  us  with  both 
latitude  and  longitude  still  affected  by  possible  errors  in  the 
D.  R.  latitude. 

On  the  other  hand,  the  Sumner  calculation  gives  us  both 
a  latitude  and  a  longitude,  but  neither  belongs  to  the  ship's 
position.  They  both  belong  to  the  position  of  the  Sumner 
point,  but  they  are  free  from  the  effects  of  any  D.  R.  errors. 
They  fix  the  Sumner  point  only,  but  they  fix  it  correctly. 
Furthermore,  our  knowledge  that  the  ship  is  somewhere 
on  the  Sumner  line  is  also  a  fact,  free  from  error.  So  what 
we  learn  from  the  Sumner  method  is  sure ;  what  we  get  by 
the  older  methods  is  all  really  D.  R.  information  in  some 


NEWER  NAVIGATION  METHODS  125 

degree.  The  Sumner  method  is  independent  of  D.  R.,  an 
advantage  of  which  the  value  cannot  be  estimated  too  highly. 

Furthermore,  it  can  be  shown  mathematically  (cf.  p.  HI) 
that  a  single  observation  can  never  really  do  more  than 
determine  a  line  on  which  the  ship  must  be.  Even  a  noon- 
sight  does  no  more  than  this ;  for  in  determining  the  ship's 
latitude,  it  really  only  makes  known  a  horizontal  line  (the 
ship's  latitude  parallel)  on  the  chart.  In  other  words,  for 
a  noon-sight  the  Sumner  line  is  horizontal,  or  has  a  bearing 
of  90°.  And  it  will  always  come  out  90°,  if  a  noon-sight  is 
worked  as  a  Sumner  observation. 

But  the  principal  purpose  of  our  present  comparison  of 
the  two  methods  of  calculation  is  to  warn  the  navigator 
against  falling  into  the  error  of  imagining  the  ship  to  be  at 
the  Sumner  point.  The  observation  does  no  more  than  tell 
us  where  the  Sumner  point  is,  and  that  the  ship  is  somewhere 
on  the  line ;  so  far  as  the  observation  is  concerned,  all  points 
on  the  line  are  equally  likely  to  be  the  ship's  true  position. 
Therefore  it  is  misleading  to  call  the  Sumner  point  the  ship's 
''most  probable  position."  Were  it  so,  a  second  observation, 
made  later  in  the  day,  would  give  another  ''most  probable 
position"  of  the  ship.  We  should  then  be  naturally  led  to 
take  as  the  ship's  final  location  a  point  midway  between  the 
two  "most  probables,"  ascribing  their  divergence  to  possible 
errors  of  observation.  But  the  ship's  real  position  we  already 
know  (p.  Ill)  to  be  at  the  intersection  of  the  two  Sumner 
lines  resulting  from  the  two  observations.  And  this  inter- 
secting point  may  be  many  miles  from  both  "most  proba- 
bles," and  from  the  above-mentioned  midpoint  between 
them. 

Less  than  two  observations  cannot  fix  the  ship's  position 
completely;  when  two  have  been  made,  a  correct  applica- 
tion of  the  Sumner  method  requires  that  the  intersection 
point  of  two  Sumner  lines  be  determined  by  calculation. 
But  before  explaining  the  method  of  doing  this,  we  must 
describe  an  excellent  alternative  way  of  making  Sumner 


126  NAVIGATION 

calculations  such  as  we  have  given  in  the  above  examples. 
The  results  are  the  same  results  as  before,  but  they  are 
obtained  with  less  work,  and  quite  without  logarithms,  by 
means  of  special  tables  such  as  our  Table  13  (p.  292),^  which 
we  shall  call  Kelvin's  Sumner  Line  Table. 

This  table  has  a  pair  of  arguments  (p.  11),  a  and  h,  a  ap- 
pearing at  the  heads  of  the  tabular  columns,  and  b  in  the 
left-hand  column  of  each  page.  Corresponding  to  these 
two  arguments,  the  table  gives  two  angles,  K  and  Q ;  so  that 
whenever  a  and  6  are  given  we  can  find  the  corresponding 
K  and  Q;  or,  if  a  and  K  should  be  given,  we  can  find  the 
corresponding  b  and  Q. 

In  the  Sumner  problem  we  obtain,  by  preparatory  calcu- 
lation (cf.  pp.  119,  123),  the  following  data: 

Declination  of  sun  (or  star);  D.  R.  latitude;  D.  R.  longitude; 
T,  the  ship's  apparent  time  of  the  observation  for  the  sun,  or  the 
hour-angle  for  a  star ; 

and  we  wish  to  get  the  computed  altitude  and  the  azimuth. 

The  principle  on  which  Table  13  depends  is  that  the  D.  R. 
latitude  and  longitude  being  always  somewhat  uncertain, 
we  can,  if  we  choose,  change  them  by  reasonable  amounts 
before  beginning  our  calculations.  The  Sumner  point  will 
then  be  determined  by  its  distance  and  bearing  from  the 
changed  D.  R.  point,  instead  of  the  original  D.  R.  point. 
By  this  device  the  tabular  calculation  is  much  facilitated. 
The  use  of  the  table  is  easy  after  a  little  practice,  the  work 
being  divided  into  a  series  of  separate  operations.  In  de- 
scribing these  operations  we  have  used  small  subscript  num- 
bers, to  distinguish  the  several  arguments,  etc. ;  as,  for  in- 
stance, in  Operation  1  we  use  ai,  bi,  K\. 

1  These  tables  were  first  published  by  Lord  Kelvin  in  1876. 
More  extended  ones  were  recently  issued  by  Lieutenant  de  Aquino, 
of  the  Brazilian  Navy;  and  these  were  reprinted  by  the  Hydro- 
graphic  Office,  United  States  Navy,  in  1917.  Aquino  also  improved 
Kelvin's  method  of  using  his  table. 


NEWER  NAVIGATION  METHODS  127 

Operation  1,  requiring  no  interpolation.  Enter  Table  13 
with : 

Arg.  Qi  =  declination,  taken  without  regard  to  +  or  —  sign,  and  cor- 
rect to  the  nearest  whole  degree  only  ; 
Arg.    6i  =  7",  if  T  is  between  0*  and  6^ ; 

=  12''  -  T,  if  T  is  between  G''  and  12* ; 
=  T  -  12\  if  T  is  between  12*  and  18*; 
=  24*  -  T,i{  T  is  between  18*  and  24*; 

and  before  use  6i  must  be  turned  into  degrees  with 
Table  9  (p.  249).  It  need  be  correct  to  the  nearest 
degree  only.  This  proceeding  will  make  6i  always 
less  than  90°. 

Then  take  from  the  table  the  tabular  angle  Ki,  also  correct 
to  the  nearest  degree  only. 

Operation  2,  requiring  simple  interpolation.  Enter  the 
table  a  second  time  with  : 

Arg.  Gi  =  the  Ki,  obtained  in  Operation  1. 

Then,  under  this  (h,  run  down  the  X-column  until  you 
find  the  declination  (taken  without  regard  to  +  or  —  sign) ; 
so  that,  in  other  words,  K2  =  declination. 

Take  from  the  table  the  angle  Q2,  which  stands  next  to 
the  declination  K2,  and  also  the  62,  which  is  in  the  left-hand 
argument  column,  in  the  same  horizontal  line  with  the 
declination  K2  in  the  i^-column.  It  will  rarely  be  possible 
to  find  the  declination  (which  must  this  time  be  exact  to 
the  nearest  minute)  in  the  X-column ;  so  that  a  simple 
interpolation  will  be  necessary  in  getting  Q2  and  62.  An 
example  of  this  interpolation  will  be  found  on  page  129 ;  and, 
as  we  shall  see,  it  is  practically  the  only  numerical  calculation 
required  in  the  whole  problem.  The  Kelvin  method  is  very 
much  shorter  than  it  looks. 

The  angle  Q2  is  used  in  choosing  the  longitude  of  the 
"changed  D.  R.  point '%•  the  latitude  of  that  point  will  be 
found  in  Operation  3.  To  utilize  Q2  for  a  sun  observation, 
calculate  the  Greenwich  apparent  time  (G.  A.  T.)  of  the 


128  NAVIGATION 

observation,  as  on  page  102,  line  (8),  and  turn  it  into  de- 
grees with  Table  9  (page  249).     Then  : 

(1)  W.  long,  of  changed  D.  R.  point  =  G.  A.  T.  —  Q2,  if,  in  Oper- 

ation 1,  T  was  less  than  6*; 

(2)  W.  long,  of  changed  D.  R.  point  =  G.  A.  T.  -  (180°  -  Q2)  if, 

in  Operation  1,  T  was  between  6^  and  12*; 

(3)  W.  long,  of  changed  D.  R.  point  =  G.  A.  T.  -  (180°  +  Q2)  if, 

in  Operation  1,  T  was  between  12*  and  18*; 

(4)  W.  long,  of  changed  D.  R.  point  =  G.  A.  T.  -  (360°  -  Q2)  if, 

in  Operation  1,  T  was  between  18*  and  24*. 

When  the  subtractions  in  these  formulas  cannot  be  made, 
the  G.  A.  T.  may  be  increased  by  360° ;  and  when  the  west 
longitude  comes  out  greater  than  180°,  subtract  it  from  360°, 
and  call  it  east  longitude. 

In  the  case  of  a  star,  we  must  use,  in  the  above  formulas, 
the  Greenwich  hour-angle,  instead  of  the  G.  A.  T.  See 
page  105,  line  (11),  for  the  method  of  obtaining  it. 

Operation  3,  requiring  no  interpolation.  Enter  the  table 
a  third  time  with  : 

Arg.  as  =  Ki,  again  as  obtained  in  Operation  1. 

(5)  Arg.  63  =  90°  -  (62  +  changed    D.    R.   lat.),    if  latitude    and 

declination  are  of  opposite  signs,  one    +  and 
one  —  ; 

(6)  Arg.  63  =  (&2  +  changed  D.  R.  lat.)  -  90°,  if  T  was  between 

90°  and  270° ; 

(7)  Arg.  63  =  90°  -  (62  -  changed   D.   R.  lat.),  if  latitude  is  less 

than  &2 ; 

(8)  Ai'g.  63  =  90°  +  (62  -  changed    D.    R.    lat.),    if    latitude    is 

greater  than  62. 

In  choosing  among  formulas  (5)  to  (8),  give  them  pre- 
cedence in  order;  do  not  use  (7)  or  (8)  if  the  conditions 
stated  for  (5)  or  (6)  are  satisfied.  And  at  this  point,  use 
your  privilege  of  choosing  any  reasonable  changed  D.  R.  lati- 
tude for  the  ship  ;  and  choose  one  that  differs  as  little  as  pos- 
sible from  the  original  D.  R.  latitude,  and  that  yet  makes 
63  a  whole  number  of  degrees.     In  this  way,  all  further 


NEWER  NAVIGATION  METHODS  129 

interpolation  is  avoided.  Having  once  chosen  among  the 
formulas,  the  latitude  is  used  without  regard  to  +  or  — 
signs. 

To  complete  Operation  3,  having  entered  the  table  with 
the  pair  of  arguments  as  and  63,  take  out  the  tabular  K3 
and  Q3. 

K3  is  now  the  computed  altitude,  to  be  used  (p.  113)  in 
locating  the  Sumner  point  from  the  changed  D.  R.  point ; 
and  Q3  is  the  sun's  true  azimuth,  which  will  always  come 
from  the  table  less  than  90°.  If  the  ship  is  in  the  northern 
hemisphere,  this  azimuth  must  be  counted  from  the  north 
point  of  the  horizon  if,  in  Operation  3,  we  used  formulas  (6) 
or  (7) ;  or  from  the  south  point  of  the  horizon,  if  we  used 
formulas  (5)  or  (8).  With  the  ship  in  the  southern  hemi- 
sphere, interchange  the  north  and  south  points  of  the  horizon 
in  these  directions.  And  in  both  hemispheres,  the  azimuth 
will  of  course  be  counted  toward  the  east  or  west,  according 
as  the  observation  was  a  ''forenoon"  or  ''afternoon"  one 
(cf.  p.  120). 

We  shall  now  use  Table  13  for  the  example  given  on  page 
119  in  condensed  form.     We  have  (p.  127)  : 

Operation  1. 

tti  =  dec.  =  23°,  p.  119,  Hne  (1),  to  the  nearest  degree ; 

61  =  T  =  2^^  41m  3p^  p  119^  line  (4)  =  40°,  to  the  nearest 
degree ;  and,  with  ai  and  61  as  arguments.  Table  13  gives 
(p.  298)  :  Ki  =  36°,  to  the  nearest  degree. 

Operation  2. 

02=  Ki  =  36°. 
K2  =  23°  24',  p.  119,  line  (1) 

and,  with  a2  and  K2,  we  must  find  Q2  and  62.  Running  down 
the  column  headed  a  =  36°  (p.  302),  we  find : 

When  K2  =  23°    5',  Q2   =  39°  43',  bz  =  29°, 
When  K2  =  23°  51',  Q2  =  40°    0',  b2  =  30°. 

We  wish  to  interpolate  for  K2  =  23°  24',  which  is  19' 
down  from  23°  5'  toward  23°  51'.     The  whole  distance  from 


130  NAVIGATION 

23°  5'  to  23°  51'  is  46'.  Therefore  we  must  interpolate 
down  ^  of  the  whole  interval  from  Q2  =  39°  43'  to  Q2  = 
40°  0'.  The  difference  between  these  two  Q2's  is  17' ;  there- 
fore the  final  Q2,  belonging  to  K2  =  23°  24',  is  39°  43'  + 
:^  X  17'  =  39°  43'  +  7'  =  39°  50'.  Similarly,  the  difference 
between  the  two  62's  being  60',  the  final  value  of  62,  for 
K2  =  23°  24',  is  29°  +  H  X  60'  =  29°  25'.  These  two 
little  interpolations  are  practically  all  the  calculation  required 
in  the  whole  problem. 

To  find  the  longitude  of  the  changed  D.  R.  point  from  the 
above  Q2  =  39°  50',  we  take  from  page  102,  line  (8), 

Greenwich  apparent  time  of  observation,  5*  2"*  35» 

which,  by  Table  9  (p.  249)  is,  75°  39' 

We  now  use  formula  (1),  page  128,  because  T,  in  Opera- 
tion 1,  was  less  than  6*.     We  get : 

W.  long,  of  ch'd  D.  R.  pt.  =  G.  A.  T.  -  Q2  =  75°  39'  -  39°  50' 
=  35°  49'  W. 

Operation  3. 

a,  =  Ki  '=  36°. 

The  D.  R.  latitude  is  +  42°  20'  (p.  119,  line  (9)) ;  and  as 
the  declination  is  — ,  we  choose  formula  (5),  page  128. 
This,  without  changing  the  D.  R.  latitude,  would  give  63  = 
90°-  (62-f  D.  R.  lat.)  =90°-  (29°  25'+ 42°  20')  =90° -71°  45' ; 
but  by  choosing  a  changed  D.  R.  latitude  of  42°  35',  we  shall 
make  63  a  whole  number  of  degrees.  So  we  have : 
63  =  90°  -  (62 -h  changed  D.  R.  latitude)  =  90°  -  (29°  25' 
-f  42°  350  =  90°  -  72°  =  18°. 

Now  we  enter  the  table  with  the  arguments  0^3  =  36°,  and 
63  =  18°,  and  obtain,  without  interpolation  (p.  302) : 

Kz  =  computed  altitude  =  14°  29', 
Qz  =  sun's  true  azimuth  =37°  22'. 

This  azimuth  must  be  counted  from  the  south  point  of 
the  horizon,  since  we  used  formula  (5)  in  Operation  3 ;  and 


NEWER  NAVIGATION  METHODS  131 

as  the  observation  was  an  afternoon  one,  the  correct  azi- 
muth will  be  S.  37°  22'  W.  (cf.  p.  19).  Counted  in  the  United 
States  Navy  way,  from  the  north  toward  the  east,  and  so 
around  to  360°,  the  azimuth  will  be  217°  22'. 

On  page  119,  we  found  :  Computed  altitude,  14°  26' ;  azi- 
muth, 217°. 

This  computed  altitude  differs  by  3'  from  the  value  just 
found  by  Table  13.  The  difference  is  due  to  our  having 
changed  the  D.  R.  point. 

From  the  changed  D.  R.  point,  in  latitude  42°  35'  N. ; 
longitude  35°  49'  W.,  we  now  calculate  (see  Condensed  Form, 
next  page)  the  position  of  the  Sumner  point  to  be :  latitude 
42°  34'  N. ;  longitude  35°  50'  W.  The  former  position,  as 
obtained  on  page  119,  was :  latitude  42°  17'  N. ;  longitude 
35°  19'  W. 

These  two  Sumner  point  positions  should  lie  on  the 
same  Sumner  line  if  the  method  of  Table  13  gives  correct 
results;  and  they  will  satisfy  this  test,  if  the  bearing 
of  a  line  joining  them  agrees  with  the  azimuth  of  the 
Sumner  line,  which  is  217°  +  90°  =  307°.  From  the  two 
Sumner  point  positions  we  have :  latitude  difference  =  17' ; 
longitude  difference  =  31' ;  departure  (Table  2,  p.  174) 
=  23.0.  The  traverse  table  (p.  164)  gives,  for  latitude  17, 
departure  23.0,  the  distance  28,  course  307°.  The  agree- 
ment is  perfect,  and  shows  that  the  same  Sumner  line 
passes  through  both  points,  though  they  are  28  miles 
apart.  This  test  also  shows  that  the  calculation  may 
indicate  any  point  on  the  Sumner  line  as  the  Sumner  point, 
if  the  D.  R.  position  of  the  ship  is  uncertain :  and  so 
we  again  call  attention  to  the  error  of  taking  the  cal- 
culated Sumner  point  as  the  ship's  most  probable  position 
(cf.  p.  125). 

We  now,  as  usual,  repeat  the  above  calculation  by  Table  13, 
in  condensed  form,  and  including  the  final  determination 
of  the  position  of  the  Sumner  point  from  the  changed  D.  R. 
point. 


132  NAVIGATION 

SUMNER  LINE  BY  TABLE  13,  CONDENSED  FORM.    SUN 

[The  following  is  taken  from  page  119.] 

Dec!.,  4h :  -  23°  23'.7  Eq.  of  time :  +  S*"  22».3 

H.  D.  :  0.1  H.  D. :  1.2 

DecL,  4^59'":  -23    24  Eq.  time:       +3     21.1 

Watch  time :  2>^   29'"  58«                        Obs'd  alt. :  14°   19' 

C.  -  W. :  2  27  8  Index :  +  4 
Chr.  time :  4  57  6  Table  6 :  +12 
Chr.  corr'n :  +28  Table  7 :  -  5 
G.  M.  T. :  4  59  14  Corr'd  alt. :  14  30 
Eq.  of  time  :  +3  21  D.  R.  lat. :  42°  20'  N. 
G.  app.  time :  5       2     35                          D.  R.  long. :  35°   16'  W. 

D.  R.  long. :  2     21       4  W.  (3) 
Ship's  app.  time,  T :     2     41     31  (4) 

[The  following  is  calculated  with  Table  13.] 

Operation  1  Operation  2 

ai  =  dec.    =  23°  02  =     Ki   =  36° 

bi  =  T       =    2h  41"*  31»(4)  K2  =  dec.   =  23°  24' 

=  40°  Table  13,  Q2  =  39°  50' 

Table  13,  Ki  =  36°  Table  13,  bz  =  29°  25' 

Greenwich  app.  time  =  5^^  2"*  35*  =  75°  39' 
By  page  128,  form.  (1),  W.  long,  of  changed  D.  R.  pt.  =  G.  A.  T.  -  Q, 

=  35°  49'  W. 
Lat.  of  changed  D.  R.  pt.   =  42°  35'  N. 

Operation  3 

a,  =  Ki  =  36° 

bi  =  90°  -  (&2  +  changed  D.  R.  lat.)    =  18° 
Table  13,  K3  =  comp'd  alt.  =  14°    29' 

Table  13,  Q3  =  azimuth  of  sun  =  37°    22' 

or,  bv  U.  S.  Navy  =  217°  22' 

Azimuth  of  Sumner  line  =  217°  22'  +  90° 

=  307°  22' 
Dist.  of  Sumner  pt.  from  changed 

D.  R.  pt.  =  corr'd  obs'd  alt.   —  comp'd  alt.  =    1'  or  1  mile 

Bearing  of  Sumner  pt.  from  changed  D.  R.  pt.   =  217°, 
since  comp'd  alt.  is  less  than  obs'd  alt. 

Dist.  1,  on  course  217°,  gives  lat.  diff.,  0'.8 ;   dep.,  0.6;  long,  diff.,  0'.8 
Lat.  of  Sumner  pt.       =  lat.  of  ch'd  D.  R.  pt.   -  lat.  diff.      =  42°  34'  N. 
Long,  of  Sumner  pt.     =  long,  of  ch'd  D.  R.  pt.  +  long.  diff.  =  35°  50'  W. 

A  practised  navigator  can  make  the  above  complete  calcu- 
lation in  a  few  minutes,  as  there  are  no  logs  used ;  and  any- 
one can  easily  obtain  the  necessary  practice  at  sea  by  simply 
forming  the  habit  of  working  his  sights  both  as  time-sights 
and  as  Sumners.  To  illustrate  the  subject  further,  we  now 
give,  in  condensed  form,  the  Star  Example  of  p.  123,  worked 
by  Table  13. 


NEWER  NAVIGATION  METHODS  133 

SUMNER  LINE  BY  TABLE  13,  CONDENSED  FORM.    STAR 

[The  following  is  taken  from  page  123.] 


Watch  time : 

16^  29"» 

48* 

Obs'd  alt.: 

40° 

3' 

C.   -W. : 

-  1     23 

50 

Index : 

+  5 

Chr.  time : 

15       5 

58 

Table  6  : 

-  1 

Chr.  corr'n : 

-  2 

28 

Table  7  : 

-  5 

G.  M.  T. : 

15       3 

30 

Corr'd  obs'd  alt. : 

40 

2 

R.  A.  mean  sun  : 

17     42 

10 

Corr'n,  past  noon : 

2 

28 

Dec.  of  Sirius : 

-16 

36 

Greenwich  sid.  time ; 

:    8     48 

8 

D.  R.  lat. : 

-35 

20 

R.  A.  of  Sirius  : 

6     41 

34 

Green,  hour-angle  : 

2       6 

34 

D.  R.  long. : 

1     22 

44  E. 

T: 

3     29 

18 

[The 

;  following  is  calculated  with  Table  13.] 

Operation 

1 

Operation  2 

a,  =  dec.    =  17° 

Os 

=    Kx  =  49° 

hx=  T         =     ?>h 

29"»  18* 

K2 

=  dec.   =  16°  36' 

=  52° 

Table  13,  Qj    =51°  57' 

Table  13,  Ki   =  49° 

Table  13,  6j     =  25°  49' 

By  page  128,  form.  (1), 
W.  long,  of  changed  D.  R.  pt.   =  Green,  hour-angle  —  Qi^ 

339°  41' 
20°     19'  E. 
Lat.  of  changed  D.  R.  pt.  =   -  35°     49' 

Operation  3 

at  =  Ki   =  49° 
By  form.  (8),  page  128,  63     =  90°  +  (62  -  changed  D.  R.  lat.)    =  80° 
Table  13,  Kt  =  comp'd  alt.  =  40°  15' 

Table  13,   Qa   =  az.  of  Sirius  =  N.    81°  25'  W. 

or,  by  U.  S.  Navy   =        278°  35' 
Az.  of  Sumner  line  =        368°  35',  or  8°  35' 

Dist.  of  Sumner  pt.  from  changed 

D.  R.  pt.  =  corr'd  obs'd  alt.   —  comp'd  alt.  =  —  13'  or  13  miles 

Bearing  of  Sumner  pt.  from  changed  D.  R.  pt.       =  99°, 

since  comp'd  alt.  is  greater  than  obs'd  alt. 

Dist.  13,  on  course  99°,  gives  lat.  diff.,  2'.0;    dep.,  12.8;  long,  diff.,  15'.9 

Lat.  of  Sumner  pt.     =  lat.  of  ch'd  D.  R.  pt.  +  lat.  diff.  =  -  35°  51' 

Long,  of  Sumner  pt.  =  long,  of  ch'd  D.  R.  pt.  +  long.  diff.   =      20°  35'  E. 

To  complete  this  part  of  our  subject,  it  remains  to  show 
how  the  position  of  the  ship  can  be  found  at  the  intersec- 
tion of  two  Sumner  lines  (pp.  Ill,  125)  resulting  from 
two  different  observations.  Figure  18  explains  the  nature  of 
the  problem ;   and  it  is  almost  exactly  the  same  figure  and 

^  Q2  being  larger  than  the  Greenwich  hour-angle,  the  latter  was 
increased  by  360°,  to  make  the  subtraction  possible  (p.  128). 


134 


NAVIGATION 


problem  treated  in  Chapter  V,  when  we  discussed  fixing  a 

ship's   position   by   means   of    ''bearings   from   the   bow" 

(p.  54). 
The  two  Sumner  lines  in  Fig.  18  are  SL  and  S'L,  passing 

through  the  two  Sumner  points  S  and  S',  whose  latitudes 

and  longitudes  are  known 
by  calculation  from  the 
observed  altitudes.  The 
bearings  or  azimuths  of  the 
two  Sumner  lines  from  the 
north  are  the  two  angles 
NSL  and  N'S'L,  which  are 
also  known  from  the  pre- 
vious calculations.  It  is 
now  required  to  find  the 
latitude  and  longitude  of 
the  intersection  point  L, 
where  the  ship  is  situated. 
The  similarity  of  this 
problem  to  the  former  one 
,  in  Chapter  V  becomes  plain, 

Fig.  18. -Intersection  of  Sumner  Lines.  '^  ^^  ^^^^^^^^  ^  ^^^^^^  ^^^P 

sailing  from  one  Sumner 
point  to  the  other,  as  from  S  to  S\  and  taking  bearings 
from  her  bow  upon  our  ship,  located  at  L.  These  bearings 
will  be  the  two  angles  S'SL  and  S"S'L.  If  the  second 
of  these  angles  should  happen  to  be  just  twice  as  big 
as  the  first,  the  distance  S'L  between  the  two  ships  at 
the  time  of  the  second  bearing  would  be  equal  (p.  54)  to 
the  distance  *S*S'  run  by  the  imagined  ship  between  the  two 
observations. 

This  would  enable  us  to  fix  the  position  of  the  imagined 
ship  at  *S',  if  L  were  a  lighthouse  ashore.  But  if  L  is  our 
ship,  and  *S'  a  Sumner  point  of  known  position,  the  same 
observations  of  bow  bearings  would  fix  the  position  of  our 
ship  at  L.     Nor  is  it  necessary  (or  possible)  to  measure 


NEWER  NAVIGATION  METHODS  135 

such  imaginary  bearings,  or  read  the  patent  log  to  get  the 
distance  run  by  an  imagined  ship. 

For  the  distance  and  bearing  of  the  second  Sumner  point 
from  the  first  can  be  obtained  from  their  known  latitudes 
and  longitudes  with  the  traverse  table.  Thus  the  line  SS' 
(marked  'distance")  and  the  bearing  (or  course)  angle 
NSS'  become  known.  Furthermore,  the  ''bow  bearing"  at 
S  is  the  angle  S'SL,  and  it  is  equal  to  the  difference  NSL  — 
NSS'.  We  have  just  seen  that  NSS'  is  obtained  from  the 
traverse  table ;  and  NSL  is  the  calculated  azimuth  of  the 
Sumner  line  through  S.  In  a  similar  way  we  get  the  other 
''bow  bearing"  S'^S'L.  If  this  were  twice  the  first  one,  the 
"required  distance"  S'L  in  the  figure  would  be  equal  to  the 
known  distance  SS'  between  the  two  Sumner  points.  If 
not,  it  can  be  easily  shown  mathematically  that : 

(1)  Required  distance  =  known  distance  X  a  factor, 

(2)  log  factor  =  sin  S'SL  -  sin  {S^'S'L  -  S'SL). 

By  these  simple  formulas  the  required  distance  S'L  might 
be  found  :  and  as  we  also  know  the  latitude  and  longitude 
of  the  Sumner  point  S',  and  the  azimuth  or  bearing  of  S'L, 
the  traverse  table  will  make  known  the  latitude  and  longi- 
tude of  the  ship  at  L.  It  is  to  be  noted  also  that  as  we  are 
at  liberty  to  call  either  of  the  Sumner  points  S',  it  is  desirable 
to  call  that  one  S'  which  has  the  larger  "bow  bearing," 
so  that  there  will  be  no  difficulty  about  subtracting  S'SL 
from  S"S'L. 

The  factor  of  formula  (2)  above  can  practically  always 
be  found  in  our  Table  14,  the  Sumner  Intersection  Table, 
without  using  logarithms.  The  pair  of  arguments  of  the 
table  are  the  smaller  "bow  bearing"  and  the  larger  "bow 
bearing";  the  tabular  number  is  the  factor  of  formula  (1) 
above,  and  will  always  give  the  distance  of  the  intersection 
point  from  that  one  of  the  two  Sumner  points  for  which 
the  bow  bearing  was  the  larger. 

And  it  should  not  be  forgotten  that  the  Sumner  line  really 


136  NAVIGATION 

extends  equally  in  both  directions  (p.  124)  from  the  Sumner 
point,  whereas,  in  Fig.  18,  we  have  extended  it  mainly 
in  the  direction  of  the  intersection  point  L.  Now  the  cal- 
culated azimuth  of  any  Sumner  line  may  be  changed  180° 
at  will,  because  the  bearings  of  the  two  ends  of  the  line  from 
the  Sumner  point  differ  by  180°,  and  we  may  take  the  bear- 
ing of  the  line  to  be  the  bearing  of  either  end  from  the  Sumner 
point  in  the  middle  of  the  line.  Figure  18  shows,  however, 
that  for  the  purpose  of  the  present  problem  we  must  choose 
the  bearing  of  that  end  of  the  line  which  is  nearest  the  point 
of  intersection  L;  nor  does  the  choice  ever  offer  difficulty, 
because  the  known  D.  R.  position  of  the  ship  at  L,  when 
compared  with  the  known  positions  of  the  two  Sumner 
points,  will  always  indicate  whether  L  bears  east  or  west 
of  either  Sumner  point,  and  also  whether  it  bears  north  or 
south.  And  the  bearing  of  L  once  chosen,  we  can  always 
find  either  of  the  two  bow  bearings  by  this  formula : 

(3)  Bow  bearing  =  bearing  of  Sumner  line  minus  bearing 
of  the  second  Sumner  point  S'  from  the  first  point  S. 

In  using  formula  (3)  it  is  allowable  to  increase  the  bear- 
ings of  the  Sumner  lines  by  360°;  when  necessary  to  make 
the  subtractions  possible,  and  if  the  formula  brings  out  bow 
bearings  larger  than  180°,  subtract  them  from  360°,  and 
proceed  as  before. 

It  is  also  always  desirable  to  draw  a  rough  sketch  for 
every  intersection  problem  occurring  on  shipboard  so  as  to 
guard  against  accidental  large  errors  like  90°  or  180°  in  ob- 
taining the  two  bow  bearings;  and  also  to  make  sure  that 
the  latitude  and  longitude  of  the  intersection  point  L  are 
correctly  computed  with  the  traverse  table. 

The  foregoing  assumes  that  the  ship  did  not  move  from 
the  point  L  between  the  two  sextant  observations  from  which 
the  two  Sumner  lines  were  calculated.  This  will  rarely 
be  the  case,  because  it  is  very  desirable  that  the  two  observa- 
tions, if  they  are  both  sun  observations,  be  separated  by 


NEWER  NAVIGATION  METHODS  137 

three  or  four  hours,  if  possible.  The  condition  of  an  unmov- 
ing  ship  will  occur  only  if  she  is  a  sailing  vessel  becalmed, 
or  a  steamer  at  anchor ;  or  if  the  two  observations  are  made 
at  nearly  the  same  time  upon  two  different  heavenly  bodies, 
such  as  two  stars. 

High  accuracy  in  the  resulting  ''fix"  (p.  53)  of  the  ship 
will  then  be  attained,  if  the  azimuths  of  the  two  stars  differ 
by  about  90°  at  the  time  of  observation.  The  same  favor- 
able condition  will  be  secured  if  one  of  the  observations  is 
made  upon  a  star  near  upper  transit  (pp.  89,  96),  in  the 
twilight  just  before  sunrise  or  after  sunset;  and  the  other 
observation,  at  nearly  the  same  time,  upon  the  sun,  when 
it  is  about  12°  or  15°  above  the  horizon. 

But  if  the  ship  has  traveled  a  considerable  distance  between 
the  two  observations,  it  is  necessary  to  allow  for  such  travel 
before  calculating  the  intersection  point.  Suppose  she  has 
gone  a  distance  D,  upon  a  course  (7,  by  D.  R.,  between  the 
two  observations.  Then  simply  find  from  Tables  1  and  2 
the  difference  of  latitude  and  longitude  corresponding  to 
distance  D  and  course  C ;  and  apply  them  as  corrections  to 
the  latitude  and  longitude  of  the  Sumner  point  belonging 
to  the  first  observation.  Everything  else,  including  the 
bearing  of  the  first  Sumner  line,  remaining  unchanged, 
the  calculation  then  proceeds  by  Table  14,  just  as  if  the 
ship  had  not  moved.  The  computed  intersection  point  is 
then  the  ship's  position  at  the  time  of  the  second  sextant 
observation. 

We  shall  now  work  some  intersection  examples. 

Suppose  we  have  two  Sumner  lines,  as  shown  in  the  rough 
sketch,  Fig.  19,  taken  on  board  a  ship  becalmed.  The 
two  sextant  observations  give : 

For  One  Sumner  Point,  S  For  the  Other  Point,  S' 

lat.^:     42°34'N.  42°     50' N. 

long. :     35°  50'  W.  35°     36'  W. 

bearing  of  Sumner  line :  307°  93°  (changed  to  273°) 

1  As  found  on  page  132. 


138 


NAVIGATION 


The  rough  sketch,  Fig.  19,  having  been  made,  and  the 
two  *'bow  bearings"  marked  with  httle  circular  arcs  as 
shown,  we  call  that  one  of  the  two  Sumner  points  S\  which 
has  the  larger  bow  bearing ;  and,  for  the  point  S',  we  change 


Fig.  19.  —  Rough  Sketch  of  Sumner  Intersection. 

the  bearing  of  the  Sumner  line  from  93°  to  180°  +  93°  = 
273°,  so  as  to  count  the  bearing  for  that  end  of  the  line  which 
is  toward  the  intersection  point  L  (p.  136).  The  other 
bearing,  307°,  for  the  point  S,  is  already  correctly  counted. 

We  now  have,  from  the  two  Sumner  point  latitudes  and 
longitudes  :  latitude  difference  =  16' ;  longitude  difference  = 
14';  departure  (Table  2,  p.  174,  for  middle  latitude  43°)  = 
10.2;  and,  for  latitude  difference  =  16,  departure  =  10.2, 
we  find  (Table  1,  p.  162),  distance  =  19,  course  =  32°.  The 
distance  between  the  two  Sumner  points  is  therefore  19 
miles,  and  the  bearing  of  S'  from  S  is  32°. 

Now  we  apply  formula  (3),  page  136,  and  find  : 

Smaller  bow  bearing  at  S  =  307°  -  32°  =  275°. 
Larger  bow  bearing  at   S'  =  273°  -  32°  =  241°. 

Being  larger  than  180°,  these  must  be  subtracted  from 
360°  (p.  136),  giving : 

Smaller  bow  bearing  =  85°;    Larger  bow  bearing  =  119°. 

Next  we  refer  to  Table  14,  and  find  with  the  smaller 
bearing  85°,  and  the  larger  119°  the  factor  1.78  (p.  322). 


NEWER  NAVIGATION  METHODS 


139 


According  to  formula  (1),  page  135,  we  then  have: 

Required  distance  LS'  =  distance  SS'  X  factor 
=  19X  1.78  =  33.8  miles. 

Therefore  the  position  of  the  ship  at  L  is  distant  33.8 
miles  from  S' ,  and  she  bears  273°.  With  this  distance  and 
bearing  or  course  angle,  the  traverse  table  (p.  154)  gives : 
latitude  =  1.8,  departure  =  33.8.  For  the  departure  33.8, 
Table  2  gives,  for  the  middle  latitude  43°  (p.  174),  differ- 
ence longitude  =  46\2.  The  bearing  273°  showing  that  the 
intersection  point  L  is  N.  and  W.  of  *S',  we  have : 

Latitude     of  ship  at  L  =  42°  50'  N.  +    r.8  =  42°  51^.8  N. 
Longitude  of  ship  at  L  =  35°  36'  W.  +  46'.2  =  36°  22'  W. 

As  a  second  example  take  the  following  two  Sumner  lines, 
as  shown  in  the  rough  sketch,  Fig.  20.  The  two  sextant 
observations  give : 


For  One  Sumner  Point,  S 

lat. :    14°  26'  N. 
long. :   77°     8'  W. 
bearing  of  line  :   53° 


For  the  Other  Point,  S' 
15°  30'     N. 
76°  22'.5  W. 
135° 


And  suppose  the  ship,  in  the  interval 
between  the  two  sextant  observations,  has 
traveled  a  distance  D  =  31  miles,  on  course 
C  =  205°.  We  must  begin  (p.  137)  by 
shifting  the  first  Sumner  point  S  sl  dis- 
tance D,  on  the  course  C.  For  this  course 
and  distance,  we  have  (Table  1,  p.  160) : 
lat.,  28M;  dep.,  13.1;  diff.  long.,  13'.5 
(Table  2,  p.  168). 

Therefore,  the  latitude  and  longitude  of 
the  first  Sumner  point  must  be  corrected  (p.  137)  as  follows  : 

For  the  point  S,  lat.  =  14°  26'  N.  -  28'.1  =  13°  58'  N. 

long.  =  77°    8'  W.  +  13'.5  =  77°  21'.5  W. 

Bearing  (unchanged)   =  53°. 
We  now  have,  for  the  two  Sumner  points :  lat.  diff.,  92' ; 


Fig.  20.  — Rough 
Sketch'of  Sumner 
Intersection, 


140  NAVIGATION 

long,  diff.,  59' ;  dep.,  57.0  (p.  169) ;  dist.,  108  miles  (p.  162) ; 
bearing  of  S'  from  *S,  32°. 

Now  we  have,  by  formula  (3),  page  136 : 

Smaller  bow  bearing  at  5  =    53°  -  32°  =   21°. 
Larger  bow  bearing  at  S'  =  135°  -  32°  =  103°. 

Table  14  (p.  319)  gives  the  factor  0.36 ;  so  that  the  ship  at 
L  is  distant  from  S'  108  X  .36  =  38.9  miles,  and  bears  135°. 
For  this  distance  and  bearing  we  have  (Table  1,  p.  166), 
latitude  =  27'. 6 ;  departure  =  27.6 ;  and  longitude  differ- 
ence (Table  2,  p.  168)  =  28'.6.  Finally,  then,  at  the  time 
of  the  second  sextant  observation,  the  ship  at  L  was  in 
latitude  15°  30'  N.  -  27'.6  =  15°  2'.4  N. ;  and  in  longitude 
76°  22'.5  W.  -  28'.6  =  75°  54'  W. 


CHAPTER  X 
A  NAVIGATOR'S   DAY  AT   SEA 

The  present  chapter  contains  a  number  of  examples  by 
means  of  which  the  reader  can  gain  facihty  in  the  use  of  the 
methods  set  forth  in  the  preceding  pages. 

The  steam  yacht  Nav  is  bound  from  New  York  to 
Colon,  and  the  captain  plans  to  take  his  departure  from 
the  Sandy  Hook  Lightship,  on  Dec.  18,  1917,  as  early  as 
possible  in  the  morning. 

The  first  bit  of  navigation,  to  be  accomplished  before  the 
yacht  leaves  her  anchorage  in  the  "Horseshoe,"  is  to  ascer- 
tain by  D.  R.  methods  the  proper  course  to  steer  from 
Sandy  Hook.  A  glance  at  the  track  chart  of  the  north 
Atlantic  shows  that  she  must  go  by  way  of  Crooked  Island 
Passage,  and  the  Windward  Passage  between  Cuba  and 
Haiti.  It  is  also  apparent  from  the  chart  that  the  first  land 
to  be  sighted  among  the  islands  is  Watlings  Island,  and  that 
the  proper  course  should  pass  to  the  eastward  of  it. 

The  position  of  Sandy  Hook  Lightship  i  is  lat.  40°  28'  N. ; 
long.  73°  50'  W.  Hinchinbroke  Rock,  at  the  southern  end 
of  Wathngs  Island,  is  in  lat.  23°  57'  N. ;  long.  74°  28'  W. 
But  the  course  should  be  shaped  for  a  point  about  12  miles 
east  of  Watlings  Island,  to  be  perfectly  safe.  The  position 
of  such  a  point  is  (approximately)  lat.  23°  57'  N. ;  long.  74° 
15'  W.2 

1  There  is  an  excellent  list  of  latitudes  and  longitudes  in  Bow- 
ditch's  "Navigator." 

2  The  difference  between  this  longitude  and  that  of  Hinchinbroke 
Rock  is  13' ;  but  13'  here  corresponds  to  about  12  miles,  on  account 
of  Table  2. 

/  141 


142  NAVIGATION 

ABSTRACT  OF   LOG.    Steam  Yacht  iVar,  Dec.  18,  1917 


Patent 

Compass 

True 

Log 

Course 

Course 

7 :  02  A.M. 

Took  departure  from  Sandy 

Hook  Lightship 

26.2 

S. 

188° 

7:21 

Sunrise,  observed  azimuth 

^1.0 

S. 

188° 

8:00 

41.0 

S. 

188° 

9:00 

57.2 

S. 

188° 

9:36 

Bow  bearing,  Barnegat .... 

67.0 

s. 

188° 

9:42 

Altitude  and  azimuth 

69.1 

s. 

188° 

9:57 

Beam  bearing,  Barnegat . . . 
(fix,  lat.  39°  45'  N.;   long. 
73°  59'  W.) 

72.5 

s. 

188° 

10:00 

73.4 

s. 

188° 

10:07 

Changed  course 

75.3 

S.|E. 

182° 

11:00 

88.7 

S.^E. 

182° 

11:42 

Ex-mer.  obs'n  lat.  39°  19' ; 

D.  R.  long.  73°  58' 

98.5 

S.^E. 

182° 

12:00 

102.6 

S.|E. 

182° 

1 :  00  P.M. 

117.7 

S.^E. 

182° 

2:00 

133.0 

S.^E. 

182° 

3:00 

' 

149.0 

S.^E. 

182° 

4:00 

163.8 

S.^E. 

182° 

4:12 

Alt.  and  az.,  fix,  lat.  38°  11' ; 

long.  73°  54' 

166.9 

S.§E. 

182° 

5:00 

182.0 

S.^E. 

182° 

6:00 

197.2 

S.fE. 

i82r 

By  the  method  of  page  20,  the  course  from  Sandy  Hook 
Lightship  should  be  181°,  and  the  distance  is  990  miles. 
These  numbers,  and  all  subsequent  numbers  in  the  present 
chapter,  should  be  verified  by  the  reader. 

The  distance  being  quite  large,  it  is  well  to  check  it  by 
the  logarithmic  method,  page  33.  The  result  by  this  method 
is:  course  181°  14',  distance  991.7  miles. 

The  chart  also  shows  that  this  course  will  carry  the  yacht 
very  near  Barnegat  Light,  on  the  coast  of  New  Jersey.  The 
position  of  this  light  is  lat.  39°  46'  N. ;  long.  74°  6'  W.  The 
captain  decides  that  it  will  be  well  to  plan  passing  this  light 


A  NAVIGATOR'S  DAY  AT   SEA  143 

at  about  5  miles'  distance.  The  position  of  a  point  5  miles 
east  of  Barnegat  Light  is  lat.  39°  46'  N.,  long.  73°  59'  W.  The 
course  and  distance  to  this  point  from  Sandy  Hook  Ship 
are  189°  and  42.5  miles.  This  course  is  so  nearly  the  same 
as  the  course  to  Watlings  Island  that  the  captain  decides 
to  steer  the  189°  course. 

All  this  work  must  be  complete  before  reaching  Sandy 
Hook,  for  the  course  from  the  lightship  must  be  ready  for 
the  quartermaster  before  the  lightship  is  passed.  And 
there  is  still  more  preliminary  work.  For  the  courses  cal- 
culated above  are  true  courses  (p.  43)  and  the  quarter- 
master must  have  the  compass  course,  so  that  he  may  be 
able  to  steer  the  yacht.  The  method  of  calculating  the 
compass  course  from  the  true  course  is  given  on  page  48  ;  and 
in  applying  it  the  captain  must  have  his  deviation  tables 
at  hand.  We  shall  assume  that  the  tables  printed  on  pages  48 
and  49  were  the  ones  furnished  by  the  compass  adjuster  for 
the  present  voyage. 

An  examination  of  the  Atlantic  track  chart  shows  that  in 
the  vicinity  of  Sandy  Hook,  the  variation,  F,  is  10°  W.,  or 
—  10°.  By  formula  (3)  (p.  49),  we  then  have,  since  the  true 
course  T  is  189°  : 

Magnetic  course  =  M=T-V  =  189°  -  (-  10°)  =  199°. 

The  second  deviation  table  (p.  49)  shows  that  when  the 
magnetic  course  (or  magnetic  bearing  of  ship's  head)  is  199°, 
the  deviation,  D,  is  +  18°.  Then,  with  V  =  -  10°,  D  =  18°, 
formula  (1),  page  45,  gives : 

Compass  error  =  ^  =  7  +  D  =  -  10°  +  18°  =  +  8^ 

And  from  formula  (2),  page  45  : 

Compass  course  C=T-  E  =  189°  -  8°  =  181° ; 

and  so  the  yacht  must  be  steered  on  a  181°  compass  course 
for  Barnegat.  But  the  quartermaster  is  to  steer  by  "  points  " 
so  that  the  course  nearest  the  181°  course  is  due  south.  The 
captain  decides  to  have  the  yacht  steered  due   south  by 


144  NAVIGATION 

compass,  and  is  prepared  to  give  the  quartermaster  his 
orders  as  soon  as  Sandy  Hook  Lightship  shall  be  reached. 

The  foregoing  preliminary  work  having  been  completed 
the  previous  day,  the  anchor  is  tripped  at  the  Horseshoe 
about  an  hour  before  daylight  on  Dec.  18,  the  weather  being 
fine,  sea  smooth,  and  wind  light  from  the  northwest.  The 
lightship  is  reached  and  passed  at  7  :  02  a.m.,  ship's  time,  civil 
reckoning,  the  ship  then  taking  her  departure.  At  that 
moment,  the  patent  log  is  read,  and  found  to  register  26.2 
miles.  The  quartermaster  gets  his  orders  to  steer  south ; 
and  all  the  above  facts  are  duly  recorded  in  the  log-book. 
And  at  every  hour  thereafter,  8,  9,  10,  etc.,  a  similar  record 
must  be  made  in  the  log-book. 

The  next  event  is  sunrise,  which  occurs  at  7  :  21,  very 
soon  after  leaving  the  lightship.  The  sun's  compass  bearing 
can  then  be  very  conveniently  observed,  and  will  furnish 
an  excellent  check  on  the  compass  adjuster.  This  observa- 
tion was  made  at  7  :  21  a.m.,  ship's  time,  civil  reckoning, 
corresponding  to  19*  21^",  Dec.  17,  ship's  apparent  time, 
astronomic  reckoning ;  and  the  sun's  bearing  or  azimuth 
was  113°  by  compass.  This  was  entered  in  the  log-book, 
and  at  the  same  time  the  patent  log  was  read,  and  found  to 
be  31.0  miles. 

To  check  the  deviation  table,  the  procedure  was  then  as 
follows : 

By  patent  log  the  yacht  had  proceeded  from  the  light- 
ship a  distance  of  31.0  —  26.2  =  4.8  miles,  on  a  compass 
course  of  180°,  or  true  course  of  188° ;  by  D.  R.,  she  had 
therefore  reached  the  position  lat.  40°  23'  N. ;  long.  73°  51'  W. 
The  sun's  declination,  from  the  almanac,  is  —  23°  23',  and 
the  (approximate  0  T  (p.  100)  is  19*  21"*.  The  sun's  true 
azimuth  is  found  from  Table  11  to  be  121°  ;  and  in  using  the 
table  for  this  purpose  take  the  altitude  of  the  sun,  for  the 

1  If  there  is  any  chance  of  this  T  being  much  in  error,  the  cap- 
tain's watch,  by  which  the  observation  is  timed,  must  be  compared 
with  the  chronometer.     See  p.  94. 


A  NAVIGATOR'S  DAY  AT  SEA  145 

moment  of  sunrise,  to  be  0°.  The  observed  compass  azi- 
muth having  been  113°,  formula  (2),  page  45,  gave  E  =  T—C 
=  121°  -  113°  =  +8°.  Then  from  formula  (1),  page  45, 
D  =E  -V  =  -\-S°-  {-  10°)  =  +  18°.  As  expected,  this 
deviation  agrees  with  the  deviation  table,  which  would 
not  be  likely  to  go  wrong  so  soon  after  the  beginning  of  a 
voyage. 

At  8  A.M.  the  patent  log  read  41.0;  and  at  9  a.m.,  57.2. 
The  course  was  still  S.  by  compass,  or  188°,  true  course. 

At  9 :  24  Barnegat  Light  was  sighted  by  the  lookout,  and 
the  mate  was  ordered  to  take  bow-and-beam  bearings  (p.  55) 
upon  it. 

At  9  :  36,  the  light  bore  225°  by  compass,  or  45°  from  the 
bow ;  patent  log,  67.0. 

At  9*  42"*  28*  by  his  watch  the  captain  took  the  altitude 
of  the  sun's  lower  limb  with  the  sextant,  and  found  it  to 
be  18°  51'.  Index  correction  was  +  3',  and  height  of  eye, 
15  feet.  C.  —  W.  was  4^  51"*  50* ;  and  the  chr.  correction 
by  the  rate  card  was  4",  slow.  Patent  log,  69.1.  At  9  :  45 
by  the  watch,  the  sun's  azimuth  was  again  observed  with 
pelorus,  and  found  to  be  137°,  compass  bearing.  It  was 
intended  to  work  a  Sumner  line  from  the  altitude  by  Kelvin's 
table;  and  the  pelorus  observation  was  made  because  the 
sun's  true  azimuth  always  comes  out  as  a  by-product,  when 
Kelvin's  table  is  used,  and  so  it  is  just  as  well  to  have  an- 
other check  on  the  deviation  table.  This  is  the  peculiar 
advantage  of  Kelvin's  table.  Without  any  additional  cal- 
culations, the  compass  is  always  checked  up  on  the  very 
course  the  ship  is  steering.  This  is  just  what  the  good 
navigator  wants. 

The  observations  could  not  be  worked  up  at  once,  be- 
cause the  captain  wished  to  see  the  result  of  the  mate's 
bow-and-beam  bearings.  At  9  :  57  by  the  watch,  Barnegat 
bore  abeam,  on  the  starboard  hand,  or  270°  by  compass,  the 
yacht  being  still  on  the  180°  compass  course.  Patent  log 
now  72.5. 

£ 


146  NAVIGATION 

Between  the  bow-and-beam  bearings  the  run  by  log  was 
72.5  —  67  =  5.5  miles.  Therefore  the  yacht  is  now  5.5 
miles  from  Barnegat  Light,  and  the  compass  bearing  of  the 
light  is  270°.  The  compass  error  being  +  8°,  the  true  bear- 
ing of  the  light  is  278° ;  and  the  bearing  of  the  yacht  from 
the  light  is  the  former  bearing  reversed,  or  278°  —  180°  =  98°, 
true.  From  this  comes  an  accurate  and  complete  position 
of  the  yacht.  Barnegat  Light  is  in  lat.  39°  46'  N. ;  long.  74°  6' 
W.  The  yacht,  5.5  miles  away  on  the  bearing  98°,  must,  by 
traverse  table,  be  in  lat.  39°  45'  N. ;  long.  73°  59'  W. 

At  10  A.M.,  the  log  was  73.4,  course  188°,  true. 

Now  the  captain  prepared  to  shape  a  new  course  to  be 
followed  from  the  Barnegat  bow-and-beam  bearing  ''fix"  in 
the  above  lat.  39°  45'  N. ;  long.  73°  59'  W.,  at  9  :  57. 

Allowing  ten  minutes  to  work  up  the  new  course,  the 
captain  plans  to  change  course  at  10 :  07.  At  that  time 
the  ship,  on  her  course  of  188°,  will  be  (at  15-knot  speed) 
2'. 5  S.  and  practically  0'  W.  of  the  Barnegat  position.  So 
the  course  will  be  changed  when  the  yacht  is  in  lat.  39°  42'  N. ; 
long.  73°  59'  W.,  at  10 :  07.  The  course  and  distance  from 
there  to  the  point  12  miles  east  of  Hinchinbroke  Rock  are : 
distance,  945  miles ;  course,  181°,  true,  or  173°  by  compass. 

Therefore,  by  the  table  on  page  52,  the  quartermaster  gets 
the  new  course  S.|E.  by  compass,  at  10  :  07.  This  corre- 
sponds to  174°  by  compass,  or  182°  true  course ;  and  at 
10 :  07,  when  the  course  was  changed,  the  patent  log  read 
75.3. 

Now  the  Sumner  line,  from  the  observation  at  9^*  42^  28' 
by  the  watch,  was  worked  by  Kelvin's  table ;  and  the  result 
was : 

Sumner  point  is  in  lat.  39°  50'  N. ;  long.  73°  56'  W. ;  bearing  of 
Sumner  line  237°. 

It  is  necessary,  as  a  check,  to  ascertain  whether  this  Sum- 
ner line  passes  through  the  position  obtained  for  the  ship 
by  the  Barnegat  bearings.  Before  doing  this,  the  Sumner 
point  must  be  shifted  by  the  method  of  page  137,  to  allow  for 


A  NAVIGATOR'S  DAY   AT   SEA  147 

the  motion  of  the  yacht  between  9 :  42,  when  the  sextant 
observation  was  made,  and  9 :  57,  when  Barnegat  bore 
abeam.  The  difference  is  15  minutes,  and  in  that  time  the 
ship  moved  south  3.4  miles  by  the  patent  log  and  an  in- 
significant distance  west. 

Therefore  the  corrected  Sumner  data  are  : 

Sumner  point  is  in  lat.  39°  46'.6  N. ;  long.  73°  56'  W. ;  bearing  of 
Sumner  line  237°. 

If  everything  fits,  this  Sumner  line  must  pass  through  the 
Barnegat  ''fix"  of  the  yacht  in  lat.  39°  45'  N. ;  long.  73°  59' 
W.,  because  the  yacht  must  have  been  somewhere  on  the 
line. 

The  traverse  table  shows  that  the  bearing  of  a  line  passing 
the  Sumner  point  and  the  yacht's  position  is  235°,  differing 
only  2°  from  the  Sumner  line  bearing ;  so  this  check  is  satis- 
factory. But  a  better  way  to  check  this  matter  is  to  deter- 
mine the  yacht's  position  from  the  intersection  of  two  lines, 
one  of  which  is  the  Sumner  line,  and  the  other  the  beam  bear- 
ing of  Barnegat  Light.  This  can  be  done  by  the  method  of 
page  133.     The  data  of  the  problem  are  : 

Sumner  point :    lat.  39°  46'.6  N. 

long.  73°  56'  W. 
Line  bears  237° 
Barnegat  Light :  lat.  39°  46'  N. 

long.  74°    6'  W. 
Line  bears  98° 

We  shall  call  Barnegat  Light  S' ;  and  then  formula  (3), 
page  136,  gives,  for  the  two  bow  bearings : 

At  Sumner  point,  S,  237°  -  266°  =    29°. 
At  Barnegat,         S\   98°  -  266°  =  168°. 

For  these  two  bearings.  Table  14  gives  the  factor  0.74,  and 
the  yacht  is  placed  6  miles  from  Barnegat,  on  the  98°  bear- 
ing. The  bow-and-beam  observations  gave  5.5  miles,  so 
the  check  by  the  Sumner  line  is  excellent. 

It  remains  for  the  captain  to  utilize  the  azimuth  observa- 


148  NAVIGATION 

tion  made  at  9 :  45.  The  bearing  of  the  Sumner  Hne  was 
237°,  and  therefore  the  sun's  true  azimuth  was  147°.  The 
observed  azimuth,  by  pelorus  (p.  145),  was  137°.  The  com- 
pass error  was  therefore  +  10°.  The  variation  being  —  10°, 
the  deviation  by  formula  (1),  page  45,  is  Z)  =  10°  —  (  -  10°)  = 
+  20°. 

On  page  143  we  found  that  the  deviation  table  made  this 
deviation  +  18° ;  so  that  the  table  appears  to  require  a 
correction  of  +2°.  The  captain  decides  not  to  correct 
the  table  for  the  present,  unless  later  azimuth  observations 
shall  confirm  it,  especially  as  the  sunrise  observation  showed 
the  adjuster's  results  to  be  correct.  Azimuth  observa- 
tions made  when  the  sun  is  high  in  the  sky  are  not  quite 
as  reliable  as  sunrise  ones.  Moreover,  the  observation  was 
made  at  9 :  45,  whereas  the  altitude  observation,  for  which 
the  true  azimuth  was  calculated  with  Kelvin's  table,  was 
made  at  9 :  42,  so  that  the  true  azimuth  must  have  been  in 
error  by  the  sun's  azimuth  change  in  three  minutes.  This 
could  have  been  avoided  by  giving  the  mate  orders  to  ob- 
serve the  azimuth  at  about  the  same  moment  when  the 
captain  took  the  altitude.  Or,  the  sun's  azimuth  change 
in  three  minutes  might  be  taken  from  the  azimuth  table,  and 
the  computed  true  azimuth  duly  corrected. 

At  11  the  log  read  88.7,  and  the  course  was  S.|E.  by  com- 
pass, or  182°,  true. 

At  about  11  :  30,  the  weather  showing  signs  of  becoming 
thick,  no  preparations  were  made  for  a  noon-sight  by  the 
method  of  page  86 ;  and  rather  than  take  the  risk  of  losing  his 
noon  observation  altogether,  the  captain  took  an  ex-me- 
ridian altitude  at  11*  42"*  0*  by  his  watch;  log  was  98.5; 
the  sextant  reading  26°  55';  index  +  3';  height  of  eye  15 
ft. ;  C.  —  W.  was  now  4*  51*"  42* ;  and  chronometer  slow  4*. 

The  observation  was  worked  by  Kelvin's  table,  and  gave 
the  Sumner  point  in  lat.  39°  20'  N. ;  long.  73°  40'  W. ;  bearing 
of  Sumner  line  86°.  Figure  21  is  a  rough  sketch  of  this  Sumner 
line.     It  is  very  nearly  horizontal ;   had  the  observation  been 


A  NAVIGATOR'S  DAY  AT   SEA 


149 


made  at  noon  precisely,  it  would  have  been  perfectly  hori- 
zontal. 

It  would  now  have  been  possible  to  move  up  the  Sumner 
line  observed  at  9 :  42,  and  obtain  an  intersection  to  fix  the 
position  of  the  yacht. 
But  this  did  not  seem 
necessary  to  the  cap- 
tain, because  of  the 
beam  bearing  obtained 
at  Barnegat  at  9  :  57, 
which  gave  a  good  fix. 

And  the  present 
Sumner  line  being  so 
nearly  horizontal,  it  is 
not  necessary  to  know 
the  longitude  very  ac- 
curately to  obtain  an 
exact  latitude.  The 
longitude  by  D.  R.  is 
sufficient,  and  it  is  73°  58'  W.  The  difference  between 
this  longitude  and  that  of  the  Sumner  point  (73°  40')  is 
18' ;  and  the  ship  at  L  (fig.  21)  bears  180°  +  86°  =  266° 
from  the  Sumner  point.  Table  2  gives  the  dep.  14.0  for 
long.  diff.  18',  in  lat.  39°.  And  for  course  266°,  dep.  14.0, 
we  find  in  Table  1,  lat.  diff.  I'.O,  so  the  yacht's  latitude  is  1' 
less  than  that  of  the  Sumner  point,  and  is  therefore  39°  19'. 
This  happens  to  be  in  exact  accord  with  the  D.  R.  latitude, 
which  was  also  39°  19'.  This  was  perfectly  satisfactory, 
and  the  captain  decided  to  carry  this  Sumner  line  forward 
for  an  intersection,  in  case  he  should  obtain  an  observation 
in  the  afternoon. 

At  12,  the  patent  log  read  102.6,  course  S.|E.,  182°  true; 
D.  R.  lat.  39°  15';  long.  73°  58';  distance  to  Watlings  Island 
918  miles. 

Had  the  yacht  been  on  a  course  other  than  almost  due 
south,  it  would  have  been  necessary  to  set  the  watch  and  the 


Fig.  21.  —  Sumner  Line  from  ex-Meridian 
Observation. 


150  NAVIGATION 

cabin  clock  to  ship's  apparent  time.  In  fact,  some  naviga- 
tors set  their  watches  to  ship's  apparent  time  before  every 
observation  (p.  94) : 

at  1,  log  read  117.7,  misty, 
at  2,  log  read  133.0,  misty, 
at  3,  log  read  149.0  misty, 
at  4,  log  read  163.8,  clearing. 

At  4*  12'"  18'  by  the  watch,  the  weather  having  cleared, 
the  altitude  of  the  sun  was  found  to  be  4°  38' ;  index  +  4' ; 
eye  15  ft.;  C.  —  W.  4*  51"*  50';  chronometer  slow  4*;  log 
166.9.  Sun's  azimuth,  observed  by  the  mate  at  the  same 
time,  came  out  224°  by  compass. 

This  observation  was  worked  for  a  Sumner  line  by  the 
Kelvin  table,  and  gave :  , 

Position  of  Sumner  point  lat.  38°  6'  N. ;  long.  73°  49'  W. ;  bearing 
of  line  145°  ;  azimuth  of  sun  235°. 

The  Sumner  line  obtained  at  1 1*  42*"  0*  was  brought  up  to 
the  time  of  the  present  observation  by  D.  R.  (p.  137),  giving  : 

position    of  11:42  Sumner 
^  point,  after  moving  it,  lat. 

38°  12' N.;  long.  73°  43' W.; 
bearing  of  the  line  86°. 
Both  lines  were  then 
sketched,  as  shown  in  Fig. 
22.     The    point    S  is    the 

Fig.  22.  —  Rough  Sketch  of  Sumner      (moved)  Sumner  point  from 

Line  Intersection.  the   11:42    observation,  S' 

that  from  the  4:12  observation.  The  intersection  point  L  is 
the  position  of  the  ship  at  4 :  12,  and  it  came  out  (p.  134) : 
lat.  38°  ir  N. ;  long.  73°  54'  W.  The  position  brought  up 
by  D.  R.  from  11 :42  was  :  lat.  38°  11' ;  long.  74°  1' ;  so  that 
there  has  been  an  easterly  set  of  the  current,  amounting  to 
7'  of  longitude  in  4|  hours.  The  sun's  true  azimuth  at 
4 :  12  was  235°,  from  the  Kelvin  table ;  and  the  pelorus 
observation  gave  224°.     The  compass  error  was  therefore 


A  NAVIGATOR'S  DAY  AT  SEA  151 

+  11°.  The  variation  being  —  10°,  the  deviation  must 
be  Z)  =  11°  -  (  -  10°  =)  +  21°.  The  deviation  table  made 
this  deviation  +  18°,  so  that  table  seems  to  require  a  correc- 
tion of  +3°.  The  pelorus  observation  of  9  :  45  gave  a  correc- 
tion of  +  2°  for  the  deviation  table ;  and  as  this  is  now 
apparently  confirmed,  the  captain  decides  to  examine  the 
chart  again,  before  finally  shaping  course  for  the  night,  to 
see  if  the  yacht  has  not  perhaps  moved  into  a  region  where 
the  variation  is  different  from  the  Sandy  Hook  variation  so 
far  used. 

At  5  the  log  read  182.0,  course  was  still  182°  true. 

The  captain  now  prepared  to  shape  the  course  for  the 
night,  and  to  change  his  course,  if  necessary,  at  6 :  00.  His 
first  step  was  to  obtain  the  D.  R.  position  at  6  :  00,  starting 
from  the  observed  position  at  4 :  12.  This  gave  position  at 
6  :  00,  by  D.  R. :  lat.  37°  41' ;  long.  73°  55'.  The  easterly 
current  ^  of  about  2'  per  hour  set  the  yacht  farther  east  about 
3'  between  4  :  12  and  6  :  00.  Therefore  he  took  the  D.  R. 
position  at  6  :  00  to  be  lat.  37°  41' ;  long.  73°  52'.  The  posi- 
tion of  the  point  of  destination,  12  miles  east  of  Watlings 
Island,  is  still :  lat.  23°  57' ;  long.  74°  15'.  The  true  course 
and  distance  to  that  point  from  the  yacht's  6  :  00  position  is 
therefore,  by  traverse  table  :  course  181^° ;  dist.  824  miles. 

A  further  examination  of  the  track  chart  shows  that  the 
variation,  which  was  —  10°  at  Sandy  Hook,  is  now  —  8°. 
The  compass  error,  from  the  last  pelorus  observation, 
was  +  11°.  Consequently,  by  the  pelorus  observation,  the 
compass  course  for  the  night  should  be  181^°  —  11°  =  170^°, 
or  S.fE.  (see  the  Table  on  p.  52).  Furthermore,  the 
variation  being  now  —  8°  and  the  error  +  11°  makes  the 
deviation  D  =E-7=  +  ll°-(-8°)  =  +  19°.  The  com- 
pass adjuster's  deviation  of  +  18°  is  therefore  vindicated, 
and  the  compass  course  S.fE.  can  be  set  for  the  night. 

At  6  the  log  read  197.2,  course  S.fE.,  or  1821°  true. 

^  Doubtless  the  Gulf  Stream. 


152  NAVIGATION 

In  conclusion,  the  captain  of  the  Nav  hopes  he  has  been 
able  to  make  his  imagined  proceedings  clear  enough  to  help 
the  young  navigator  in  planning  his  own  first  day's  work  at 
sea.  May  it  be  the  first  of  many  happy  and  successful  days. 
And  let  him  not  forget,  when  attempting  to  verify  the 
various  calculations  and  problems  of  the  Nav,  that  every 
observation  in  this  book  has  been  prepared  by  calculation, 
and  none  is  the  result  of  actual  sextant  observing.  Should 
inconsistencies  or  errors  be  found  by  any  young  navigator,  it 
is  hoped  that  he  will  make  them  known  so  that  they  may  be 
corrected,  in  case  the  Nav  shall  be  required  to  make  another 
voyage  in  a  second  edition. 


LIST   OF   TABLES 

1.  Traverse  Table;   explained  on  pages  10  and  19;   and  its 

use  in  the  Sumner  method  on  pages  113,  135 154 

2.  Conversion   of  longitude  difference   and  departure;    ex- 

plained on  page  16 168 

3.  Number  logarithms ;   explained  on  page  23 178 

4.  Trigonometric  logarithms  ;  explained  on  page  31 196 

5.  Meridional  parts ;   explained  on  page  35 241 

6.  Sextant  Correction  Table ;   explained  on  page  72 247 

7.  Dip  correction ;   explained  on  page  73 247 

8.  Conversion  of  hours  and  minutes  into  decimals  of  a  day ; 

explained  on  page  80 248 

9.  Conversion  of  degrees  and  minutes  of  longitude  and  hours 

and  minutes  of  time 249 

10.  Haversines ;   explained  on  page  99 250 

11.  Azimuth  Table ;   explained  on  page  113 284 

12.  Auxiliary  Azimuth  Table ;   explained  on  page  115 290 

13.  Kelvin's  Sumner  Line  Table ;   explained  on  page  126 292 

14.  Sumner  Intersection  Table ;   explained  on  page  135 318 


PUBLISHERS'   NOTE 

Table  3,  Number  Logarithms,  has  been  reprinted  from  "The 
Macmillan  Logarithmic  and  Trigonometric  Tables,"  New  York, 
1917. 


153 


154 


Table  1.    Trayerse  Table 


1° 

2° 

iPt 

.  3° 

4 

0 

5 

0 

^Pt 

.  6° 

7° 

(179°,  181°, 

(178°,  182°, 

(177°, 

183°, 

(176°, 

184°, 

(175°, 

185°, 

(174°, 

186°, 

(173°,  187°, 

DiST. 

359°) 

358°) 

357°) 

356°) 

355°) 

354 

n 

353°) 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

1 

1.0 

0.0 

1.0 

0.0 

1.0 

0.1 

1.0 

0.1 

1.0 

0.1 

1.0 

0.1 

1.0 

0.1 

2 

2.0 

0.0 

2.0 

0.1 

2.0 

0.1 

2.0 

0.1 

2.0 

0.2 

2.0 

0.2 

2.0 

0.2 

3 

3.0 

0.1 

3.0 

0.1 

3.0 

0.2 

3.0 

0.2 

3.0 

0.3 

3.0 

0.3 

3.0 

0.4 

4 

4.0 

0.1 

4.0 

0.1 

4.0 

0.2 

4.0 

0.3 

4.0 

0.3 

4.0 

0.4 

4.0 

0.5 

5 

5.0 

0.1 

5.0 

0.2 

5.0 

0.3 

5.0 

0.3 

5.0 

0.4 

5.0 

0.5 

5.0 

0.6 

6 

6.0 

0.1 

6.0 

0.2 

6.0 

0.3 

6.0 

0.4 

6.0 

0.5 

6.0 

0.6 

6.0 

0.7 

7 

7.0 

0.1 

7.0 

0.2 

7.0 

0.4 

7.0 

0.5 

7.0 

0.6 

7,0 

0.7 

6.9 

0.9 

8 

8.0 

0.1 

8.0 

0.3 

8.0 

0.4 

8.0 

0.6 

8.0 

0.7 

8.0 

0.8 

7.9 

1.0 

9 

9.0 

0.2 

9.0 

0.3 

9.0 

0.5 

9.0 

0.6 

9.0 

0.8 

9.0 

0.9 

8.9 

1.1 

10 

10.0 

0.2 

10.0 

0.3 

10.0 

0.5 

10.0 

0.7 

10.0 

0.9 

9.9 

1.0 

9.9 

1.2 

11 

11.0 

0.2 

11.0 

0.4 

11.0 

0.6 

11.0 

0.8 

11.0 

1.0 

10.9 

1.1 

10.9 

1.3 

12 

12.0 

0.2 

12.0 

0.4 

12.0 

0.6 

12.0 

0.8 

12.0 

1.0 

11.9 

1.3 

11.9 

1.5 

13 

13.0 

0.2 

13.0 

0.5 

13.0 

0.7 

13.0 

0.9 

13.0 

1.1 

12.9 

1.4 

12.9 

1.6 

14 

14.0 

0.2 

14.0 

0.5 

14.0 

0.7 

14.0 

1.0 

13.9 

1.2 

13.9 

1.5 

13.9 

1.7 

15 

15.0 

0.3 

15.0 

0.5 

15.0 

0.8 

15.0 

1.0 

14.9 

1.3 

14.9 

1.6 

14.9 

1.8 

16 

16.0 

0.3 

16.0 

0.6 

16.0 

0.8 

16.0 

1.1 

15.9 

1.4 

15.9 

1.7 

15.9 

1.9 

17 

17.0 

0.3 

17.0 

0.6 

17.0 

0.9 

17.0 

1.2 

16.9 

1.5 

16.9 

1.8 

16.9 

2.1 

18 

18.0 

0.3 

18.0 

0.6 

18.0 

0.9 

18.0 

1.3 

17.9 

1.6 

17.9 

1.9 

17.9 

2.2 

19 

19.0 

0.3 

19.0 

0.7 

19.0 

1.0 

19.0 

1.3 

18.9 

1.7 

18.9 

2.0 

18.9 

2.3 

20 

20.0 

0.3 

20.0 

0.7 

20.0 

1.0 

20.0 

1.4 

19.9 

1.7 

19.9 

2.1 

19.9 

2.4 

21 

21.0 

0.4 

21.0 

0.7 

21.0 

1.1 

20:9 

1.5 

20.9 

1.8 

20.9 

2.2 

20.8 

2.6 

22 

22.0 

0.4 

22.0 

0.8 

22.0 

1.2 

21.9 

1.5 

21.9 

1.9 

21.9 

2.3 

21.8 

2.7 

23 

23.0 

0.4 

23.0 

0.8 

23.0 

1.2 

22.9 

1.6 

22.9 

2.0 

22.9 

2.4 

22.8 

2.8 

24 

24.0 

0.4 

24.0 

0.8 

24.0 

1.3 

23.9 

1.7 

23.9 

2.1 

23.9 

2.5 

23.8 

2.9 

25 

25.0 

0.4 

25.0 

0.9 

25.0 

1.3 

24.9 

1.7 

24.9 

2.2 

24.9 

2.6 

24.8 

3.0 

26 

26.0 

0.5 

26.0 

0.9 

26.0 

1.4 

25.9 

1.8 

25.9 

2.3 

25.9 

2.7 

25.8 

3.2 

27 

27.0 

0.5 

27.0 

0.9 

27.0 

1.4 

26.9 

1.9 

26.9 

2.4 

26.9 

2.8 

26.8 

3.3 

28 

28.0 

0.5 

28.0 

1.0 

28.0 

1.5 

27.9 

2.0 

27.9 

2.4 

27.8 

2.9 

27.8 

3.4 

29 

29.0 

0.5 

29.0 

1.0 

29.0 

1.5 

28.9 

2.0 

28.9 

2.5 

28.8 

3.0 

28.8 

3.5 

30 

30.0 

0.5 

30.0 

1.0 

30.0 

1.6 

29.9 

2.1 

29.9 

2.6 

29.8 

3.1 

29.8 

3.7 

31 

31.0 

0.5 

31.0 

1.1 

31.0 

1.6 

30.9 

2.2 

30.9 

2.7 

30.8 

3.2 

30.8 

3.8 

32 

32.0 

0.6 

32.0 

1.1 

32.0 

1.7 

31.9 

2.2 

31.9 

2.8 

31.8 

3.3 

31.8 

3.9 

33 

33.0 

0.6 

33.0 

1.2 

33.0 

1.7 

32.9 

2.3 

32.9 

2.9 

32.8 

3.4 

32.8 

4.0 

34 

34.0 

0.6 

34.0 

1.2 

34.0 

1.8 

33.9 

2.4 

33.9 

3.0 

33.8 

3.6 

33.7 

4.1 

35 

35.0 

0.6 

35.0 

1.2 

35.0 

1.8 

34.9 

2.4 

34.9 

3.1 

34.8 

3.7 

34.7 

4.3 

36 

36.0 

0.6 

36.0 

1.3 

36.0 

1.9 

35.9 

2.5 

35.9 

3.1 

35.8 

3.8 

35.7 

4.4 

37 

37.0 

0.6 

37.0 

1.3 

36.9 

1.9 

36.9 

2.6 

36.9 

3.2 

36.8 

3.9 

36.7 

4.5 

38 

38.0 

0.7 

38.0 

1.3 

37.9 

2.0 

37.9 

2.7 

37.9 

3.3 

37.8 

4.0 

37.7 

4.6 

39 

39.0 

0.7 

39.0 

1.4 

38.9 

2.0 

38.9 

2.7 

38.9 

3.4 

38.8 

4.1 

38.7 

4.8 

40 

40.0 

0.7 

40.0 

1.4 

39.9 

2.1 

39.9 

2.8 

39.8 

3.5 

39.8 

4.2 

39.7 

4.9 

41 

41.0 

0.7 

41.0 

1.4 

40.9 

2.1 

40.9 

2.9 

40.8 

3.6 

40.8 

4.3 

40.7 

5.0 

42 

42.0 

0.7 

42.0 

1.5 

41.9 

2.2 

41.9 

2.9 

41.8 

3.7 

41.8 

4.4 

41.7 

5.1 

43 

43.0 

0.8 

43.0 

1.5 

42.9 

2.3 

42.9 

3.0 

42.8 

3.7 

42.8 

4.5 

42.7 

5.2 

44 

44.0 

0.8 

44.0 

1.5 

43.9 

2.3 

43.9 

3.1 

43.8 

3.8 

43.8 

4.6 

43.7 

5.4 

45 

45.0 

0.8 

45.0 

1.6 

44.9 

2.4 

44.9 

3.1 

44.8 

3.9 

44.8 

4.7 

44.7 

5.5 

46 

46.0 

0.8 

46.0 

1.6 

45.9 

2.4 

45.9 

3.2 

45.8 

4.0 

45.7 

4.8 

45.7 

5.6 

47 

47.0 

0.8 

47.0 

1.6 

46.9 

2.5 

46.9 

3.3 

46.8 

4.1 

46.7 

4.9 

46.6 

5.7 

48 

48.0 

0.8 

48.0 

1.7 

47.9 

2.5 

47.9 

3.3 

47.8 

4.2 

47.7 

5.0 

47.6 

5.8 

49 

49.0 

0.9 

49.0 

1.7 

48.9 

2.6 

48.9 

3.4 

48.8 

4.3 

48.7 

5.1 

48.6 

6.0 

50 

50.0 

0.9 

50.0 

1.7 

49.9 

2.6 

49.9 

3.5 

49.8 

4.4 

49.7 

5.2 

49.6 

6.1 

100 

100.0 

1.7 

99.9 

3.5 

99.9 

5.2 

99.8 

7.0 

99.6 

8.7 

99.5 

10.5 

99.3 

12.2 

200 

200.0 

3.5 

199.9 

7.0 

199.7 

10.5 

199.5 

14.0 

199.2 

17.4 

198.9 

20.9 

198.5 

24.4 

300 

300.0 

5.2 

299.8 

10.5 

299.6 

15.7 

299.3 

20.9 

298.9 

26.1 

298.4 

31.4 

297.8 

36.6 

400 

399.9 

7.0 

399.8 

13.9 

399.4 

20.9 

399.0 

27.9 

398.5 

34.9 

397.8 

41.8 

397.0 

48.7 

500 

499.9 

8.8 

499.7 

17.4 

499.3 

26.2 

498.8 

34.8 

498.1 

43.6 

497.3 

52.3 

496.3 

61.0 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

(91°,  269°, 

(92°,  268°, 

(93°,  2 

67°, 

(94°,  2 

66°, 

(95°,  2 

65°, 

(96°,  2 

64°, 

(97°,  263°, 

271°) 

272°) 

273 

°) 

274 

°) 

275 

") 

276 

') 

277°) 

89° 

88° 

7|Pt 

87° 

86 

0 

85 

0 

7|Pt. 

84° 

83° 

Table  1.    Traverse  Table 


155 


1° 

2° 

i  Pt.  3° 

4° 

5° 

h  Pt.  6° 

7°           1 

(179°,  181°, 

(178°,  182°, 

(177°,  183°, 

(176°,  184°, 

(175°,  185°, 

(174°.  186°, 

(173' 

,187°. 

DiST. 

359°) 

358°) 

357°) 

356°) 

355°) 

354°) 

353°)         1 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

51 

51.0 

0.9 

51.0 

1.8 

50.9 

2.7 

50.9 

3.6 

50.8 

4.4 

50.7 

5.3 

50.6 

6.2 

52 

52.0 

0.9 

52.0 

1.8 

51.9 

2.7 

51.9 

3.6 

51.8 

4.5 

51.7 

5.4 

51.6 

6.3 

53 

53.0 

0.9 

53.0 

1.8 

52.9 

2.8 

52.9 

3.7 

52.8 

4.6 

52.7 

5.5 

52.6 

6.5 

54 

54.0 

0.9 

54.0 

1.9 

53.9 

2.8 

53.9 

3.8 

53.8 

4.7 

53.7 

5.6 

53.6 

6.6 

55 

55.0 

1.0 

55.0 

1.9 

54.9 

2.9 

54.9 

3.8 

54.8 

4.8 

54.7 

5.7 

54.6 

6.7 

56 

56.0 

1.0 

56.0 

2.0 

55.9 

2.9 

55.9 

3.9 

55.8 

4.9 

55.7 

5.9 

55.6 

6.8 

57 

57.0 

1.0 

57.0 

2.0 

56.9 

3.0 

56.9 

4.0 

56.8 

5.0 

56.7 

6.0 

56.6 

6.9 

58 

58.0 

1.0 

58.0 

2.0 

57.9 

3.0 

57.9 

4.0 

57.8 

5.1 

57.7 

6.1 

57.6 

7.1 

59 

59.0 

1.0 

59.0 

2.1 

58.9 

3.1 

58.9 

4.1 

58.8 

5.1 

58.7 

6.2 

58.6 

7.2 

60 

60.0 

1.0 

60.0 

2.1 

59.9 

3.1 

59.9 

4.2 

59.8 

5.2 

59.7 

6.3 

59.6 

7.3 

61 

61.0 

1.1 

61.0 

2.1 

60.9 

3.2 

60.9 

4.3 

60.8 

5.3 

60.7 

6.4 

60.5 

7.4 

62 

62.0 

1.1 

62.0 

2.2 

61.9 

3.2 

61.8 

4.3 

61.8 

5.4 

61.7 

6.5 

61.5 

7.6 

63 

63.0 

1.1 

63.0 

2.2 

62.9 

3.3 

62.8 

4.4 

62.8 

5.5 

62.7 

6.6 

62.5 

7.7 

64 

64.0 

1.1 

64.0 

2.2 

63.9 

3.3 

63.8 

4.5 

63.8 

5.6 

63.6 

6.7 

63.5 

7.8 

65 

65.0 

1.1 

65.0 

2.3 

64.9 

3.4 

64.8 

4.5 

64.8 

5.7 

64.6 

6.8 

64.5 

7.9 

66 

66.0 

1.2 

66.0 

2.3 

65.9 

3.5 

65.8 

4.6 

65.7 

5.8 

65.6 

6.9 

65.5 

8:0 

67 

67.0 

1.2 

67.0 

2.3 

66.9 

3.5 

66.8 

4.7 

66.7 

5.8 

66.6 

7.0 

66.5 

8.2 

68 

68.0 

1.2 

68.0 

2.4 

67.9 

3.6 

67.8 

4.7 

67.7 

5.9 

67.6 

7.1 

67.5 

8.3 

69 

69.0 

1.2 

69.0 

2.4 

68.9 

3.6 

68.8 

4.8 

68.7 

6.0 

68.6 

7.2 

68.5 

8.4 

70 

70.0 

1.2 

70.0 

2.4 

69.9 

3.7 

69.8 

4.9 

69.7 

6.1 

69.6 

7.3 

69.5 

8.5 

71 

71.0 

1.2 

71.0 

2.5 

70.9 

3.7 

70.8 

5.0 

70.7 

6.2 

70.6 

7.4 

70.5 

8.7 

72 

72.0 

1.3 

72.0 

2.5 

71.9 

3.8 

71.8 

5.0 

71.7 

6.3 

71.6 

7.5 

71.5 

8.8 

73 

73.0 

1.3 

73.0 

2.5 

72.9 

3.8 

72.8 

5.1 

72.7 

6.4 

72.6 

7.6 

72.5 

8.9 

74 

74.0 

1.3 

74.0 

2.6 

73.9 

3.9 

73.8 

5.2 

73.7 

6.4 

73.6 

7.7 

73.4 

9.0 

75 

75.0 

1.3 

75.0 

2.6 

74.9 

3.9 

74.8 

5.2 

74.7 

6.5 

74.6 

7.8 

74.4 

9.1 

76 

76.0 

1.3 

76.0 

2.7 

75.9 

4.0 

75.8 

5.3 

75.7 

6.6 

75.6 

7.9 

75.4 

9.3 

77 

77.0 

1.3 

77.0 

2.7 

76.9 

4.0 

76.8 

5.4 

76.7 

6.7 

76.6 

8.0 

76.4 

9.4 

78 

78.0 

1.4 

78.0 

2.7 

77.9 

4.1 

77.8 

5.4 

77.7 

6.8 

77.6 

8.2 

77.4 

9.5 

79 

79.0 

1.4 

79.0 

2.8 

78.9 

4.1 

78.8 

5.5 

78.7 

6.9 

78.6 

8.3 

78.4 

9.6 

80 

80.0 

1.4 

80.0 

2.8 

79.9 

4.2 

79.8 

5.6 

79.7 

7.0 

79.6 

8.4 

79.4 

9.7 

81 

81.0 

1.4 

81.0 

2.8 

80.9 

4.2 

80.8 

5.7 

80.7 

7.1 

80.6 

8.5 

80.4 

9.9 

82 

82.0 

1.4 

82.0 

2.9 

81.9 

4.3 

81.8 

5.7 

81.7 

7.1 

81.6 

8.6 

81.4 

10.0 

83 

83.0 

1.4 

82.9 

2.9 

82.9 

4.3 

82.8 

5.8 

82.7 

7.2 

82.5 

8.7 

82.4 

10.1 

84 

84.0 

1.5 

83.9 

2.9 

83.9 

4.4 

83.8 

5.9 

83.7 

7.3 

83.5 

8.8 

83.4 

10.2 

85 

85.0 

1.5 

84.9 

3.0 

84.9 

4.4 

84.8 

5.9 

84.7 

7.4 

84.5 

8.9 

84.4 

10.4 

86 

86.0 

1.5 

85.9 

3.0 

85.9 

4.5 

85.8 

6.0 

85.7 

7.5 

85.5 

9.0 

85.4 

10.5 

87 

87.0 

1.5 

86.9 

3.0 

86.9 

4.6 

86.8 

6.1 

86.7 

7.6 

86.5 

9.1 

86.4 

10.6 

88 

88.0 

1.5 

87.9 

3.1 

87.9 

4.6 

87.8 

6.1 

87.7 

7.7 

87.5 

9.2 

87.3 

10.7 

89 

89.0 

1.6 

88.9 

3.1 

88.9 

4.7 

88.8 

6.2 

88.7 

7.8 

88.5 

9.3 

88.3 

10.8 

90 

90.0 

1.6 

89.9 

3.1 

89.9 

4.7 

89.8 

6.3 

89.7 

7.8 

89.5 

9.4 

89.3 

11.0 

91 

91.0 

1.6 

90.9 

3.2 

90.9 

4.8 

90.8 

6.3 

90.7 

7.9 

90.5 

9.5 

90.3 

11.1 

92 

92.0 

1.6 

91.9 

3.2 

91.9 

4.8 

91.8 

6.4 

91.6 

8.0 

91.5 

9.6 

91.3 

11.2 

93 

93.0 

1.6 

92.9 

3.2 

92.9 

4.9 

92.8 

6.5 

92.6 

8.1 

92.5 

9.7 

92.3 

11.3 

94 

94.0 

1.6 

93.9 

3.3 

93.9 

4.9 

93.8 

6.6 

93.6 

8.2 

93.5 

9.8 

93.3 

11.5 

95 

95.0 

1.7 

94.9 

3.3 

94.9 

5.0 

94.8 

6.6 

94.6 

8.3 

94.5 

9.9 

94.3 

11.6 

96 

96.0 

1.7 

95.9 

3.4 

95.9 

5.0 

95.8 

6.7 

95.6 

8.4 

95.5 

10.0 

95.3 

11.7 

97 

97.0 

1.7 

96.9 

3.4 

96.9 

5.1 

96.8 

6.8 

96.6 

8.5 

96.5 

10.1 

96.3 

11.8 

,98 

98.0 

1.7 

97.9 

3.4 

97.9 

5.1 

97.8 

6.8 

97.6 

8.5 

97.5 

10.2 

97.3 

11.9 

99 

99.0 

1.7 

98.9 

3.5 

98.9 

5.2 

98.8 

6.9 

98.6 

8.6 

98.5 

10.3 

98.3 

12.1 

100 

100.0 

1.7 

99.9 

3.5 

99.9 

5.2 

99.8 

7.0 

99.6 

8.7 

99.5 

10.5 

99.3 

12.2 

600 

599.9 

10.5 

599.6 

20.9 

599.2 

31.4 

598.6 

41.9 

597.7 

52.3 

596.7 

62.7 

595.5 

73.1 

700 

699.8 

12.2 

699.5 

24.4 

699.0 

36.6 

698.2 

48.8 

697.2 

61.0 

696.1 

73.2 

694.9 

85.3 

800 

799.8 

14.0 

799.5 

27.9 

798.9 

41.9 

798.0 

55.8 

796.9 

69.7 

795.6 

83.6 

794.1 

97.5 

900 

899.7 

15.7 
Lat. 

899.3 

31.4 

898.6 

47.1 

897.6 

62.8 

896.4 

78.4 

895.0 

94.1 

893.3 

109.6 

Dep. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

(91°.  269°, 

(92°.  268° 

(93°,  267°, 

(94°.  266°, 

(95°.  265°, 

(96°,  264°, 

(97°, 

263°, 

271°) 

272°) 

273°) 

274°) 

275°) 

276°) 

27 

7°) 

89° 

88° 

7f  Pt.  87° 

86° 

85° 

7i  Pt.  84° 

8 

r 

156 


Table  1.    Traverse  Table 


f  Pt.  8° 

9° 

10° 

1  Pt.  11° 

12° 

13° 

U  Pt.  14° 

(172°,  188°, 

(171°,  189°, 

(170°,  190°, 

(169°,  191°, 

(168°,  192°, 

(167°, 

193°, 

(166°,  194°, 

DiST. 

352°) 

351°) 

350°) 

349°) 

348°) 

347°)        1 

346°) 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

1 

1.0 

0.1 

1.0 

0.2 

1.0 

0.2 

1.0 

0.2 

1.0 

0.2 

1.0 

0.2 

1.0 

0.2 

2 

2.0 

0.3 

2.0 

0.3 

2.0 

0.3 

2.0 

0.4 

2.0 

0.4 

1.9 

0.4 

1.9 

0.5 

3 

3.0 

0.4 

3.0 

0.5 

3.0 

0.5 

2.9 

0.6 

2.9 

0.6 

2.9 

0.7 

2.9 

0.7 

4 

4.0 

0.6 

4.0 

0.6 

3.9 

0.7 

3.9 

0.8 

3.9 

0.8 

3.9 

0.9 

3.9 

1.0 

5 

5.0 

0.7 

4.9 

0.8 

4.9 

0.9 

4.9 

1.0 

4.9 

1.0 

4.9 

1.1 

4.9 

1.2 

6 

5.9 

0.8 

5.9 

0.9 

5.9 

1.0 

5.9 

1.1 

5.9 

1.2 

5.8 

1.3 

5.8 

1.5 

7 

6.9 

1.0 

6.9 

1.1 

6.9 

1.2 

6.9 

1.3 

6.8 

1.5 

6.8 

1.6 

6.8 

1.7 

8 

7.9 

1.1 

7.9 

1.3 

7.9 

1.4 

7.9 

1.5 

7.8 

1.7 

7.8 

1.8 

7.8 

1.9 

9 

8.9 

1.3 

8.9 

1.4 

8.9 

1.6 

8.8 

1.7 

8.8 

1.9 

8.8 

2.0 

8.7 

2.2 

10 

9.9 

1.4 

9.9 

1.6 

9.8 

1.7 

9.8 

1.9 

9.8 

2.1 

9.7 

2.2 

9.7 

2.4 

11 

10.9 

1.5 

10.9 

1.7 

10.8 

1.9 

10.8 

2.1 

10.8 

2.3 

10.7 

2.5 

10.7 

2.7 

12 

11.9 

1.7 

11.9 

1.9 

11.8 

2.1 

11.8 

2.3 

11.7 

2.5 

11.7 

2.7 

11.6 

2.9 

13 

12.9 

1.8 

12.8 

2.0 

12.8 

2.3 

12.8 

2.5 

12.7 

2.7 

12.7 

2.9 

12.6 

3.1 

14 

13.9 

1.9 

13.8 

2.2 

13.8 

2.4 

13.7 

2.7 

13.7 

2.9 

13.6 

3.1 

13.6 

3.4 

15 

14.9 

2.1 

14.8 

2.3 

14.8 

2.6 

14.7 

2.9 

14.7 

3.1 

14.6 

3.4 

14.6 

3.6 

16 

15.8 

2.2 

15.8 

2.5 

15.8 

2.8 

15.7 

3.1 

15.7 

3.3 

15.6 

3.6 

15.5 

3.9 

17 

16.8 

2.4 

16.8 

2.7 

16.7 

3.0 

16.7 

3.2 

16.6 

3.5 

16.6 

3.8 

16.5 

4.1 

18 

17.8 

2.5 

17.8 

2.9 

17.7 

3.1 

17.7 

3.4 

17.6 

3.7 

17.5 

4.0 

17.5 

4.4 

19 

18.8 

2.6 

18.8 

3.0 

18.7 

3.3 

18.7 

3.6 

18.6 

4.0 

18.5 

4.3 

18.4 

4.6 

20 

19.8 

2.8 

19.8 

3.1 

19.7 

3.5 

19.6 

3.8 

19.6 

4.2 

19.5 

4.5 

19.4 

4.8 

21 

20.8 

2.9 

20.7 

3.3 

20.7 

3.6 

20.6 

4.0 

20.5 

4.4 

20.5 

4.7 

20.4 

5.1 

22 

21.8 

3.1 

21.7 

3.4 

21.7 

3.8 

21.6 

4.2 

21.5 

4.6 

21.4 

4.9 

21.3 

5.3 

23 

22.8 

3.2 

22.7 

3.6 

22.7 

4.0 

22.6 

4.4 

22.5 

•4.8 

22.4 

5.2 

22.3 

5.6 

24 

23.8 

3.3 

23.7 

3.8 

23.6 

4.2 

23.6 

4.6 

23.5 

5.0 

23.4 

5.4 

23.3 

5.8 

25 

24.8 

3.5 

24.7 

3.9 

24.6 

4.3 

24.5 

4.8 

24.5 

5.2 

24.4 

5.6 

24.3 

6.0 

26 

25.7 

3.6 

25.7 

4.1 

25.6 

4.5 

25.5 

5.0 

25.4 

5.4 

25.3 

5.8 

25.2 

6.3 

27 

26.7 

3.8 

26.7 

4.2 

26.6 

4.7 

26.5 

5.2 

26.4 

5.6 

26.3 

6.1 

26.2 

6.5 

28 

27.7 

3.9 

27.7 

4.4 

27.6 

4.9 

27.5 

5.3 

27.4 

5.8 

27.3 

6.3 

27.2 

6.8 

29 

28.7 

4.0 

28.6 

4.5 

28.6 

5.0 

28.5 

5.5 

28.4 

6.0 

28.3 

6.5 

28.1 

7.0 

30 

29.7 

4.2 

29.6 

4.7 

29.5 

5.2 

29.4 

5.7 

29.3 

6.2 

29.2 

6.7 

29.1 

7.3 

31 

30.7 

4.3 

30.6 

4.8 

30.5 

5.4 

30.4 

5.9 

30.3 

6.4 

30.2 

7.0 

30.1 

7.5 

32 

31.7 

4.5 

31.6 

5.0 

31.5 

5.6 

31.4 

6.1 

31.3 

6.7 

31.2 

7.2 

31.0 

7.7 

33 

32.7 

4.6 

32.6 

5.2 

32.5 

5.7 

32.4 

6.3 

32.3 

6.9 

32.2 

7.4 

32.0 

8.0 

34 

33.7 

4.7 

33.6 

5.3 

33.5 

5.9 

33.4 

6.5 

33.3 

7.1 

33.1 

7.6 

33.0 

8.2 

35 

34.7 

4.9 

34.6 

5.5 

34.5 

6.1 

34.4 

6.7 

34.2 

7.3 

34.1 

7.9 

34.0 

8.5 

36 

35.6 

5.0 

35.6 

5.6 

35.5 

6.3 

35.3 

6.9 

35.2 

7.5 

35.1 

8.1 

34.9 

8.7 

37 

36.6 

5.1 

36.5 

5.8 

36.4 

6.4 

36.3 

7.1 

36.2 

7.7 

36.1 

8.3 

35.9 

9.0 

38 

37.6 

5.3 

37.5 

5.9 

37.4 

6.6 

37.3 

7.3 

37.2 

7.9 

37.0 

8.5 

36.9 

9.2 

39 

38.6 

5.4 

38.5 

6.1 

38.4 

6.8 

38.3 

7.4 

38.1 

8.1 

38.0 

8.8 

37.8 

9.4 

40 

39.6 

5.6 

39.5 

6.3 

39.4 

6.9 

39.3 

7.6 

39.1 

8.3 

39.0 

9.0 

38.8 

9.7 

41 

40.6 

5.7 

40.5 

6.4 

40.4 

7.1 

40.2 

7.8 

40.1 

8.5 

39.9 

9.2 

39.8 

9.9 

42 

41.6 

5.8 

41.5 

6.6 

41.4 

7.3 

41.2 

8.0 

41.1 

8.7 

40.9 

9.4 

40.8 

10.2 

43 

42.6 

6.0 

42.5 

6.7 

42.3 

7.5 

42.2 

8.2 

42.1 

8.9 

41.9 

9.7 

41.7 

10.4 

44 

43.6 

6.1 

43.5 

6.9 

43.3 

7.6 

43.2 

8.4 

43.0 

9.1 

42.9 

9.9 

42.7 

10.6 

45 

44.6 

6.3 

44.4 

7.0 

44.3 

7.8 

44.2 

8.6 

44.0 

9.4 

43.8 

10.1 

43.7 

10.9 

46 

45.6 

6.4 

45.4 

7.2 

45.3 

8.0 

45.2 

8.8 

45.0 

9.6 

44.8 

10.3 

44.6 

11.1 

47 

46.5 

6.5 

46.4 

7.4 

46.3 

8.2 

46.1 

9.0 

46.0 

9.8 

45.8 

10.6 

45.6 

11.4 

48 

47.5 

6.7 

47.4 

7.5 

47.3 

8.3 

47.1 

9.2 

47.0 

10.0 

46.8 

10.8 

46.6 

11.6 

49 

48.5 

6.8 

48.4 

7.7 

48.3 

8.5 

48.1 

9.3 

47.9 

10.2 

47.7 

11.0 

47.5 

11.9 

50 

49.5 

7.0 

49.4 

7.8 

49.2 

8.7 

49.1 

9.5 

48.9 

10.4 

48.7 

11.2 

48.5 

12.1 

100 

99.0 

13.9 

98.8 

15.6 

98.5 

17.4 

98.2 

19.1 

97.8 

20.8 

97.4 

22.5 

97.0 

24.2 

200 

198.1 

27.8 

197.5 

31.3 

197.0 

34.7 

196.3 

38.2 

195.6 

41.6 

194.9 

45.0 

194.1 

48.4 

300 

297.1 

41.8 

296.3 

46.9 

295.4 

52.1 

294.5 

57.2 

293.4 

62.4 

292.3 

67.5 

291.1 

72.6 

400 

396.1 

55.7 

395.1 

62.6 

393.9 

69.5 

392.6 

76.3 

391.3 

83.1 

389.8 

90.0 

388.1 

96.7 

500 

495.1 

69.6 

493.8 

78.2 

492.4 

86.8 

490.8 

95.4 

489.1 

104.0 

487.2 

112.4 

485.1 

121.0 

Dep. 

Lat, 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

(98°,  262°, 

(99°,  261°, 

(100°,  260°, 

(101°,  259°, 

(102°,  258°, 

(103° 

257°, 

(104°,  256°, 

278°) 

279°) 

280°) 

281°) 

282°) 

28 

3°) 

284°) 

7|  Pt.  82° 

81° 

80° 

7  Pt.  79° 

78° 

7 

7° 

6f  Pt.  76° 

The  1-Pt.  or  11°  Courses  are :  N.  by  E.,  N.  by  W.,  S.  by  E.,  S.  by  W. 


Table  1.    Traverse  Table 


157 


f  Pt 

.  8° 

9 

D 

10° 

1  Pt.  11° 

12° 

13° 

UPt.  14° 

(172°, 

188°, 

(171°, 

189°. 

(170°,  190°, 

(169°,  191°, 

(168°,  192°, 

(167°,  193°, 

(166°.  194°, 

DiST. 

352°)        1 

351°)        1 

350°) 

349°) 

348°) 

347°) 

346°) 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

51 

50.5 

7.1 

50.4 

8.0 

50.2 

8.9 

50.1 

9.7 

49.9 

10.6 

49.7 

11.5 

49.5 

12.3 

52 

51.5 

7.2 

51.4 

8.1 

51.2 

9.0 

51.0 

9.9 

50.9 

10.8 

50.7 

11.7 

50.5 

12.6 

53 

52.5 

7.4 

52.3 

8.3 

52.2 

9.2 

52.0 

10.1 

51.8 

11.0 

51.6 

11.9 

51.4 

12.8 

54 

53.5 

7.5 

53.3 

8.4 

53.2 

9.4 

53.0 

10.3 

52.8 

11.2 

52.6 

12.1 

52.4 

13.1 

55 

54.5 

7.7 

54.3 

8.6 

54.2 

9.6 

54.0 

10.5 

53.8 

11.4 

53.6 

12.4 

53.4 

13.3 

56 

55.5 

7.8 

55.3 

8.8 

55.1 

9.7 

55.0 

10.7 

54.8 

11.6 

54.6 

12.6 

54.3 

13.5 

57 

56.4 

7.9 

56.3 

8.9 

56.1 

9.9 

56.0 

10.9 

55.8 

11.9 

55.5 

12.8 

55.3 

13.8 

58 

57.4 

8.1 

57.3 

9.1 

57.1 

10.1 

56.9 

11.1 

56.7 

12.1 

56.5 

13.0 

56.3 

14.0 

59 

58.4 

8.2 

58.3 

9.2 

58.1 

10.2 

57.9 

11.3 

57.7 

12.3 

57.5 

13.3 

57.2 

14.3 

60 

59.4 

8.4 

59.3 

9.4 

59.1 

10.4 

58.9 

11.4 

58.7 

12.5 

58.5 

13.5 

58.2 

14.5 

61 

60.4 

8.5 

60.2 

9.5 

60.1 

10.6 

59.9 

11.6 

59.7 

12.7 

59.4 

13.7 

59.2 

14.8 

62 

61.4 

8.6 

61.2 

9.7 

61.1 

10.8 

60.9 

11.8 

60.6 

12.9 

60.4 

13.9 

60.2 

15.0 

63 

62.4 

8.8 

62.2 

9.9 

62.0 

10.9 

61.8 

12.0 

61.6 

13.1 

01.4 

14.2 

61.1 

15.2 

64 

63.4 

8.9 

63.2 

10.0 

63.0 

11.1 

62.8 

12.2 

62.6 

13.3 

62.4 

14.4 

62.1 

15.5 

65 

64.4 

9.0 

64.2 

10.2 

64.0 

11.3 

63.8 

12.4 

63.6 

13.5 

63.3 

14.6 

63.1 

15.7 

66 

65.4 

9.2 

65.2 

10.3 

65.0 

11.5 

64.8 

12.6 

64.6 

13.7 

64.3 

14.8 

64.0 

16.0 

67 

66.3 

9.3 

66.2 

10.5 

66.0 

11.6 

65.8 

12.8 

65.5 

13.9 

65.3 

15.1 

65.0 

16.2 

68 

67.3 

9.5 

67.2 

10.6 

67.0 

11.8 

66.8 

13.0 

66.5 

14.1 

66.3 

15.3 

66.0 

16.5 

69 

68.3 

9.6 

68.2 

10.8 

68.0 

12.0 

67.7 

13.2 

67.5 

14.3 

67.2 

15.5 

67.0 

16.7 

70 

69.3 

9.7 

69.1 

11.0 

68.9 

12.2 

68.7 

13.4 

68.5 

14.6 

68.2 

15.7 

67.9 

16.9 

71 

70.3 

9.9 

70.1 

11.1 

69.9 

12.3 

69.7 

13.5 

69.4 

14.8 

69.2 

16.0 

68.9 

17.2 

72 

71.3 

10.0 

71.1 

11.3 

70.9 

12.5 

70.7 

13.7 

70.4 

15.0 

70.2 

16.2 

69.9 

17.4 

73 

72.3 

10.2 

72.1 

11.4 

71.9 

12.7 

71.7 

13.9 

71.4 

15.2 

71.1 

16.4 

70.8 

17.7 

74 

73.3 

10.3 

73.1 

11.6 

72.9 

12.8 

72.6 

14.1 

72.4 

15.4 

72.1 

16.6 

71.8 

17.9 

75 

74.3 

10.4 

74.1 

11.7 

73.9 

13.0 

73.6 

14.3 

73.4 

15.6 

73.1 

16.9 

72.8 

18.1 

76 

75.3 

10.6 

75.1 

11.9 

74.8 

13.2 

74.6 

14.5 

74.3 

15.8 

74.1 

17.1 

73.7 

18.4 

77 

76.3 

10.7 

76.1 

12.0 

75.8 

13.4 

75.6 

14.7 

75.3 

16.0 

75.0 

17.3 

74.7 

18.6 

78 

77.2 

10.9 

77.0 

12.2 

76.8 

13.5 

76.6 

14.9 

76.3 

16.2 

76.0 

17.5 

75.7 

18.9 

79 

78.2 

11.0 

78.0 

12.4 

77.8 

13.7 

77.5 

15.1 

77.3 

16.4 

77.0 

17.8 

76.7 

19.1 

80 

79.2 

11.1 

79.0 

12.5 

78.8 

13.9 

78.5 

15.3 

78.3 

16.6 

77.9 

18.0 

77.6 

19.4 

81 

80.2 

11.3 

80.0 

12.7 

79.8 

14.1 

79.5 

15.5 

79.2 

16.8 

78.9 

18.2 

78.6 

19.6 

82 

81.2 

11.4 

81.0 

12.8 

80.8 

14.2 

80.5 

15.6 

80.2 

17.0 

79.9 

18.4 

79.6 

19.8 

83 

82.2 

11.6 

82.0 

13.0 

81.7 

14.4 

81.5 

15.8 

81.2 

17.3 

80.9 

18.7 

80.5 

20.1 

84 

83.2 

11.7 

83.0 

13.1 

82.7 

14.6 

82.5 

16.0 

82.2 

17.5 

81.8 

18.9 

81.5 

20.3 

85 

84.2 

11.8 

84.0 

13.3 

83.7 

14.8 

83.4 

16.2 

83.1 

17.7 

82.8 

19.1 

82.5 

20.6 

86 

85.2 

12.0 

84.9 

13.5 

84.7 

14.9 

84.4 

16.4 

84.1 

17.9 

83.8 

19.3 

83.4 

20.8 

87 

86.2 

12.1 

85.9 

13.6 

85.7 

15.1 

85.4 

16.6 

85.1 

18.1 

84.8 

19.6 

84.4 

21.0 

88 

87.1 

12.2 

86.9 

13.8 

86.7 

15.3 

86.4 

16.8 

86.1 

18.3 

85.7 

19.8 

85.4 

21.3 

89 

88.1 

12.4 

87.9 

13.9 

87.6 

15.5 

87.4 

17.0 

87.1 

18.5 

86.7 

20.0 

86.4 

21.5 

90 

89.1 

12.5 

88.9 

14.1 

88.6 

15.6 

88.3 

17.2 

88.0 

18.7 

87.7 

20.2 

87.3 

21.8 

91 

90.1 

12.7 

89.9 

14.2 

89.6 

15.8 

89.3 

17.4 

89.0 

18.9 

88.7 

20.5 

88.3 

22.0 

92 

91.1 

12.8 

90.9 

14.4 

90.6 

16.0 

90.3 

17.6 

90.0 

19.1 

89.6 

20.7 

89.3 

22.3 

93 

92.1 

12.9 

91.9 

14.5 

91.6 

16.1 

91.3 

17.7 

91.0 

19.3 

90.6 

20.9 

90.2 

22.5 

94 

93.1 

13.1 

92.8 

14.7 

92.6 

16.3 

92.3 

17.9 

91.9 

19.5 

91.6 

21.1 

91.2 

22.7 

95 

94.1 

13.2 

93.8 

14.9 

93.6 

16.5 

93.3 

18.1 

92.9 

19.8 

92.6 

21.4 

92.2 

23.0 

96 

95.1 

13.4 

94.8 

15.0 

94.5 

16.7 

94.2 

18.3 

93.9 

20.0 

93.5 

21.6 

93.1 

23.2 

97 

96.1 

13.5 

95.8 

15.2 

95.5 

16.8 

95.2 

18.5 

94.9 

20.2 

94.5 

21.8 

94.1 

23.5 

98 

97.0 

13.6 

96.8 

15.3 

96.5 

17.0 

96.2 

18.7 

95.9 

20.4 

95.5 

22.0 

95.1 

23.7 

99 

98.0 

13.8 

97.8 

15.5 

97.5 

17.2 

97.2 

18.9 

96.8 

20.6 

96.5 

22.3 

96.1 

24.0 

100 

99.0 

13.9 

98.8 

15.6 

98.5 

17.4 

98.2 

19.1 

97.8 

20.8 

97.4 

22.5 

97.0 

24.2 

600 

594.2 

83.5 

592.6 

93.8 

590.9 

104.2 

589.0 

114.5 

586.9 

124.7 

584.6 

135.0 

582.2 

145.1 

700 

693.3 

97.4 

691.3 

109.4 

689.5 

121.5 

687.1 

133.6 

684.7 

145.5 

682.1 

157.5 

679.2 

169.3 

800 

792.3 

111.4 

790.2 

125.1 

787.9 

139.0 

785.2 

152.6 

782.5 

166.3 

779.4 

180.0 

776.2 

193.6 

900 

891.3 

125.2 

888.8 

140.8 

886.3 

156.3 

883.3 

171.7 

880.2 

187.1 

876. S 
Dep. 

202.4 

873.2 

217.7 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Lat. 

Dep. 

Lat. 

(98°. 

262°, 

(99°, 

261°, 

(100°,  260°, 

(101°,  259°, 

(102°,  258°, 

(103°,  257°, 

(104°,  256°, 

27 

8°) 

27 

9°) 

280°) 

281°) 

282°) 

283°) 

284°) 

7\  P 

t.  82° 

8 

1° 

80° 

7  Pt.  79° 

78° 

77° 

6f  Pt.  76° 

The  7-Pt.  or  79°  Courses  are  :  E.  by  N.,  W.  by  N.,  E.  by  S.,  W.  by  S. 


158 


Table  1.    Traverse  Table 


15° 

16° 

UPt.  17° 

18° 

19° 

U  Pt.  20° 

DiST. 

(165° 

195°, 

(164°,  196°, 

(163°,  197*, 

(162°. 

198°, 

(161°, 

199°, 

(160° 

,  200°, 

345°) 

344°) 

343°) 

342°) 

341°) 

340°)         1 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

1 

1.0 

0.3 

1.0 

0.3 

1.0 

0.3 

1.0 

0.3 

0.9 

0.3 

0.9 

0.3 

2 

1.9 

0.5 

1.9 

0.6 

1.9 

0.6 

1.9 

0.6 

1.9 

0.7 

1.9 

0.7 

3 

2.9 

0.8 

2.9 

0.8 

2.9 

0.9 

2.9 

0.9 

2.8 

1.0 

2.8 

1.0 

4 

3.9 

1.0 

3.8 

1.1 

3.8 

1.2 

3.8 

1.2 

3.8 

1.3 

3.8 

1.4 

5 

4.8 

1.3 

4.8 

1.4 

4.8 

1.5 

4.8 

1.5 

4.7 

1.6 

4.7 

1.7 

6 

5.8 

1.6 

5.8 

1.7 

5.7 

1.8 

5.7 

1.9 

5.7 

2.0 

5.6 

2.1 

7 

6.8 

1.8 

6.7 

1.9 

6.7 

2.0 

6.7 

2.2 

6.6 

2.3 

6.6 

2.4 

8 

7.7 

2.1 

7.7 

2.2 

7.7 

2.3 

7.6 

2.5 

7.6 

2.6 

7.5 

2.7 

9 

8.7 

2.3 

8.7 

2.5 

8.6 

2.6 

8.6 

2.8 

8.5 

2.9 

8.5 

3.1 

10 

9.7 

2.6 

9.6 

2.8 

9.6 

2.9 

9.5 

3.1 

9.5 

3.3 

9.4 

3.4 

11 

10.6 

2.8 

10.6 

3.0 

10.5 

3.2 

10.5 

3.4 

10.4 

3.6 

10.3 

3.8 

12 

11.6 

3.1 

11.5 

3.3 

11.5 

3.5 

11.4 

3.7 

11.3 

3.9 

11.3 

4.1 

13 

12.6 

3.4 

12.5 

3.6 

12.4 

3.8 

12.4 

4.0 

12.3 

4.2 

12.2 

4.4 

14 

13.5 

3.6 

13.5 

3.9 

13.4 

4.1 

13.3 

4.3 

13.2 

4.6 

13.2 

4.8 

15 

14.5 

3.9 

14.4 

4.1 

14.3 

4.4 

14.3 

4.6 

14.2 

4.9 

14.1 

5.1 

16 

15.5 

4.1 

15.4 

4.4 

15.3 

4.7 

15.2 

4.9 

15.1 

5.2 

15.0 

5.5 

17 

16.4 

4.4 

16.3 

4.7 

16.3 

5.0 

16.2 

5.3 

16.1 

5.5 

16.0 

5.8 

18 

17.4 

4.7 

17.3 

5.0 

17.2 

5.3 

17.1 

5.6 

17.0 

5.9 

16.9 

6.2 

19 

18.4 

4.9 

18.3 

5.2 

18.2 

5.6 

18.1 

5.9 

18.0 

6.2 

17.9 

6.5 

20 

19.3 

5.2 

19.2 

5.5 

19.1 

5.8 

19.0 

6.2 

18.9 

6.5 

18.8 

6.8 

21 

20.3 

5.4 

20.2 

5.8 

20.1 

6.1 

20.0 

6.5 

19.9 

6.8 

19.7 

7.2 

22 

21.3 

5.7 

21.1 

6.1 

21.0 

6.4 

20.9 

6.8 

20.8 

7.2 

20.7 

7.5 

23 

22.2 

6.0 

22.1 

6.3 

22.0 

6.7 

21.9 

7.1 

21.7 

7.5 

21.6 

7.9 

24 

23.2 

6.2 

23.1 

6.6 

23.0 

7.0 

22.8 

7.4 

22.7 

7.8 

22.6 

8.2 

25 

24.1 

6.5 

24.0 

6.9 

23.9 

7.3 

23.8 

7.7 

23.6 

8.1 

23.5 

8.6 

26 

25.1 

6.7 

25.0 

7.2 

24.9 

7.6 

24.7 

8.0 

24.6 

8.5 

24.4 

8.9 

27 

26.1 

7.0 

26.0 

7.4 

25.8 

7.9 

25.7 

8.3 

25.5 

8.8 

25.4 

9.2 

28 

27.0 

7.2 

26.9 

7.7 

26.8 

8.2 

26.6 

8.7 

26.5 

9.1 

26.3 

9.6 

29 

28.0 

7.5 

27.9 

8.0 

27.7 

8.5 

27.6 

9.0 

27.4 

9.4 

27.3 

9.9 

30 

29.0 

7.8 

28.8 

8.3 

28.7 

8.8 

28.5 

9.3 

28.4 

9.8 

28.2 

10.3 

31 

29.9 

8.0 

29.8 

8.5 

29.6 

9.1 

29.5 

9.6 

29.3 

10.1 

29.1 

10.6 

32 

30.9 

8.3 

30.8 

8.8 

30.6 

9.4 

30.4 

9.9 

30.3 

10.4 

30.1 

10.9 

33 

31.9 

8.5 

31.7 

9.1 

31.6 

9.6 

31.4 

10.2 

31.2 

10.7 

31.0 

11.3 

34 

32.8 

8.8 

32.7 

9.4 

32.5 

9.9 

32.3 

10.5 

32.1 

11.1 

31.9 

11.6 

35 

33.8 

9.1 

33.6 

9.6 

33.5 

10.2 

33.3 

10.8 

33.1 

11.4 

32.9 

12.0 

36 

34.8 

9.3 

34.6 

9.9 

34.4 

10.5 

34.2 

11.1 

34.0 

11.7 

33.8 

12.3 

37 

35.7 

9.6 

35.6 

10.2 

35.4 

10.8 

35.2 

11.4 

35.0 

12.0 

34.8 

12.7 

38 

36.7 

9.8 

36.5 

10.5 

36.3 

11.1 

36.1 

11.7 

35.9 

12.4 

35.7 

13.0 

39 

37.7 

10.1 

37.5 

10.7 

37.3 

11.4 

37.1 

12.1 

36.9 

12.7 

36.6 

13.3 

40 

38.6 

10.4 

38.5 

11.0 

38.3 

11.7 

38.0 

12.4 

37.8 

13.0 

37.6 

13.7 

41 

39.6 

10.6 

39.4 

11.3 

39.2 

12.0 

39.0 

12.7 

38.8 

13.3 

38.5 

14.0 

42 

40.6 

10.9 

40.4 

11.6 

40.2 

12.3 

39.9 

13.0 

39.7 

13.7 

39.5 

14.4 

43 

41.5 

11.1 

41.3 

11.9 

41.1 

12.6 

40.9 

13.3 

40.7 

14.0 

40.4 

14.7 

44 

42.5 

11.4 

42.3 

12.1 

42.1 

12.9 

41.8 

13.6 

41.6 

14.3 

41.3 

15.0 

45 

43.5 

11.6 

43.3 

12.4 

43.0 

13.2 

42.8 

13.9 

42.5 

14.7 

42.3 

15.4 

46 

44.4 

11.9 

44.2 

12.7 

44.0 

13.4 

43.7 

14.2 

43.5 

15.0 

43.2 

15.7 

47 

45.4 

12.2 

45.2 

13.0 

44.9 

13.7 

44.7 

14.5 

44.4 

15.3 

44.2 

16.1 

48 

46.4 

12.4 

46.1 

13.2 

45.9 

14.0 

45.7 

14.8 

45.4 

15.6 

45.1 

16.4 

49 

47.3 

12.7 

47.1 

13.5 

46.9 

14.3 

46.6 

15.1 

46.3 

16.0 

46.0 

16.8 

50 

48.3 

12.9 

48.1 

13.8 

47.8 

14.6 

47.6 

15.5 

47.3 

16.3 

47.0 

17.1 

100 

96.6 

25.9 

96.1 

27.6 

95.6 

29.2 

95.1 

30.9 

94.6 

32.6 

94.0 

34.2 

200 

193.2 

51.8 

192.3 

55.1 

191.3 

58.5 

190.2 

61.8 

189.1 

65.1 

187.9 

68.4 

300 

289.8 

77.6 

288.4 

82.7 

286.9 

87.7 

285.3 

92.7 

283.7 

97.7 

281.9 

102.6 

400 

386.3 

103.5 

384.5 

110.2 

382.5 

117.0 

380.4 

123.6 

378.2 

130.2 

375.9 

136.8 

500 

483.0 

129.4 

480.6 

137.8 

478.1 

146.2 

475.5 

154.5 

472.8 

162.8 

469.9 

171.0 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

(105° 

255°, 

(106°,  254°, 

(107°,  253°, 

(108° 

252°, 

(109° 

251°, 

(110° 

,  250°, 

28 

5°) 

286°) 

287°) 

28 

8°) 

28 

9°) 

2C 

»0°) 

7 

5°        1        74° 

6^  Pt.  73° 

7J 

2° 

7] 

L° 

6iP 

t.  70° 

Table  1.    Traverse  Table 


169 


15° 

16° 

H  Pt.  17° 

18° 

19° 

If  Pt.  20° 

(165°,  195°, 

(164° 

,  196°, 

(163°,  197°, 

(162°,  198°, 

(161= 

,  199°, 

(160°,  200°, 

DiST. 

345°) 

344°) 

343°) 

342°) 

341°) 

340°) 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

51 

49.3 

13.2 

49.0 

14.1 

48.8 

14.9 

48.5 

15.8 

48.2 

16.6 

47.9 

17.4 

52 

50.2 

13.5 

50.0 

14.3 

49.7 

15.2 

49.5 

16.1 

49.2 

16.9 

48.9 

17.8 

53 

51.2 

13.7 

50.9 

14.6 

50.7 

15.5 

50.4 

16.4 

50.1 

17.3 

49.8 

18.1 

54 

52.2 

14.0 

51.9 

14.9 

51.6 

15.8 

51.4 

16.7 

51.1 

17.6 

50.7 

18.5 

55 

53.1 

14.2 

52.9 

15.2 

52.6 

16.1 

52.3 

17.0 

52.0 

17.9 

51.7 

18.8 

56 

54.1 

14.5 

53.8 

15.4 

53.6 

16.4 

53.3 

17.3 

52.9 

18.2 

52.6 

19.2 

57 

55.1 

14.8 

54.8 

15.7 

54.5 

16.7 

54.2 

17.6 

53.9 

18.6 

53.6 

19.5 

58 

56.0 

15.0 

55.8 

16.0 

55.5 

17.0 

55.2 

17.9 

54.8 

18.9 

54.5 

19.8 

50 

57.0 

15.3 

56.7 

16.3 

56.4 

17.2 

56.1 

18.2 

55.8 

19.2 

55.4 

20.2 

60 

58.0 

15.5 

57.7 

16.5 

57.4 

17.5 

57.1 

18.5 

56.7 

19.5 

56.4 

20.5 

61 

58.9 

15.8 

58.6 

16.8 

58.3 

17.8 

58.0 

18.9 

57.7 

19.9 

57.3 

20.9 

62 

59.9 

16.0 

59.6 

17.1 

59.3 

18.1 

59.0 

19.2 

58.6 

20.2 

58.3 

21.2 

63 

60.9 

16.3 

60.6 

17.4 

60.2 

18.4 

59.9 

19.5 

59.6 

20.5 

59.2 

21.5 

64 

61.8 

16.6 

61.5 

17.6 

61.2 

18.7 

60.9 

19.8 

60.5 

20.8 

60.1 

21.9 

65 

62.8 

16.8 

62.5 

17.9 

62.2 

19.0 

61.8 

20.1 

61.5 

21.2 

61.1 

22.2 

66 

63.8 

17.1 

63.4 

18.2 

63.1 

19.3 

62.8 

20.4 

62.4 

21.5 

62.0 

22.6 

67 

64.7 

17.3 

64.4 

18.5 

64.1 

19.6 

63.7 

20.7 

63.3 

21.8 

63.0 

22.9 

68 

65.7 

17.6 

65.4 

18.7 

65.0 

19.9 

64.7 

21.0 

64.3 

22.1 

63.9 

23.3 

69 

66.6 

17.9 

66.3 

19.0 

66.0 

20.2 

65.6 

21.3 

65.2 

22.5 

64.8 

23.6 

70 

67.6 

18.1 

67.3 

19.3 

66.9 

20.5 

66.6 

21.6 

66.2 

22.8 

65.8 

23.9 

71 

68.6 

18.4 

68.2 

19.6 

67.9 

20.8 

67.5 

21.9 

67.1 

23.1 

66.7 

24.3 

72 

69.5 

18.6 

69.2 

19.8 

68.9 

21.1 

68.5 

22.2 

68.1 

23.4 

67.7 

24.6 

73 

70.5 

18.9 

70.2 

20.1 

69.8 

21.3 

69.4 

22.6 

69.0 

23.8 

68.6 

25.0 

74 

71.5 

19.2 

71.1 

20.4 

70.8 

21.6 

70.4 

22.9 

70.0 

24.1 

69.5 

25.3 

75 

72.4 

19.4 

72.1 

20.7 

71.7 

21.9 

71.3 

23.2 

70.9 

24.4 

70.5 

25.7 

76 

73.4 

19.7 

73.1 

20.9 

72.7 

22.2 

72.3 

23.5 

71.9 

24.7 

71.4 

26.0 

77 

74.4 

19.9 

74.0 

21.2 

73.6 

22.5 

73.2 

23.8 

72.8 

25.1 

72.4 

26.3 

78 

75.3 

20.2 

75.0 

21.5 

74.6 

22.8 

74.2 

24.1 

73.8 

25.4 

73.3 

26.7 

79 

76.3 

20.4 

75.9 

21.8 

75.5 

23.1 

75.1 

24.4 

74.7 

25.7 

74.2 

27.0 

80 

77.3 

20.7 

76.9 

22.1 

76.5 

23.4 

76.1 

24.7 

75.6 

26.0 

75.2 

27.4 

81 

78.2 

21.0 

77.9 

22.3 

77.5 

23.7 

77.0 

25.0 

76.6 

26.4 

76.1 

27.7 

82 

79.2 

21.2 

78.8 

22.6 

78.4 

24.0 

78.0 

25.3 

77.5 

26.7 

77.1 

28.0 

83 

80.2 

21.5 

79.8 

22.9 

79.4 

24.3 

78.9 

25.6 

78.5 

27.0 

78.0 

28.4 

84 

81.1 

21.7 

80.7 

23.2 

80.3 

24.6 

79.9 

26.0 

79.4 

27.3 

78.9 

28.7 

85 

82.1 

22.0 

81.7 

23.4 

81.3 

24.9 

80.8 

26.3 

80.4 

27.7 

79.9 

29.1 

86 

83.1 

22.3 

82.7 

23.7 

82.2 

25.1 

81.8 

26.6 

81.3 

28.0 

80.8 

29.4 

87 

84.0 

22.5 

83.6 

24.0 

83.2 

25.4 

82.7 

26.9 

82.3 

28.3 

81.8 

29.8 

88 

85.0 

22.8 

84.6 

24.3 

84.2 

25.7 

83.7 

27.2 

83.2 

28.7 

82.7 

30.1 

89 

86.0 

23.0 

85.6 

24.5 

85.1 

26.0 

84.6 

27.5 

84.2 

29.0 

83.6 

30.4 

90 

86.9 

23.3 

86.5 

24.8 

86.1 

26.3 

85.6 

27.8 

85.1 

29.3 

84.6 

30.8 

91 

87.9 

23.6 

87.5 

25.1 

87.0 

26.6 

86.5 

28.1 

86.0 

29.6 

85.5 

31.1 

92 

88.9 

23.8 

88.4 

25.4 

88.0 

26.9 

87.5 

28.4 

87.0 

30.0 

86.5 

31.5 

93 

89.8 

24.1 

89.4 

25.6 

88.9 

27.2 

88.4 

28.7 

87.9 

30.3 

87.4 

31.8 

94 

90.8 

24.3 

90.4 

25.9 

89.9 

27.5 

89.4 

29.0 

88.9 

30.6 

88.3 

32.1 

95 

91.8 

24.6 

91.3 

26.2 

90.8 

27.8 

90.4 

29.4 

89.8 

30.9 

89.3 

32.5 

96 

92.7 

24.8 

92.3 

26.5 

91.8 

28.1 

91.3 

29.7 

90.8 

31.3 

90.2 

32.8 

97 

93.7 

25.1 

93.2 

26.7 

92.8 

28.4 

92.3 

30.0 

91.7 

31.6 

91.2 

33.2 

98 

94.7 

25.4 

94.2 

27.0 

93.7 

28.7 

93.2 

30.3 

92.7 

31.9 

92.1 

33.5 

99 

95.6 

25.6 

95.2 

27.3 

94.7 

28.9 

94.2 

30.6 

93.6 

32.2 

93.0 

33.9 

100 

96.6 

25.9 

96.1 

27.6 

95.6 

29.2 

95.1 

30.9 

94.6 

32.6 

94.0 

34.2 

600 

579.5 

155.3 

576.8 

165.4 

573.8 

175.4 

570.6 

185.4 

567.3 

195.3 

563.8 

205.2 

700 

676.1 

181.1 

672.8 

193.0 

669.4 

204.6 

665.8 

216.3 

661.9 

227.9 

657.9 

239.4 

800 

772.7 

207.0 

769.0 

220.5 

765.0 

233.9 

760.8 

247.3 

756.5 

260.4 

751.8 

273.6 

900 

869.2 

232.9 

865.0 

248.0 

860.6 

263.1 

855.9 

278.1 

850.9 

292.9 

845.7 

307.8 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

(105°,  255°, 

(106°, 

254°, 

(107°,  253°, 

(108°,  252°, 

(109° 

251°, 

(110°,  250°, 

285°) 

28 

S°) 

287°) 

288°) 

28 

9°) 

290°) 

75° 

74 

1° 

6|Pt.  73° 

72° 

7 

1° 

70° 

160 


Table  1.    Traverse  Table 


21 

o 

22°       1 

2Pt. 

23° 

24°       1 

21  Pt 

.25° 

26° 

DiST. 

(159°, 

201°, 

(158°, 

202°, 

(157°, 

203°, 

(15G°, 

204°, 

(155°, 

205°, 

(154°,  206°, 

339°)        1 

338°)        1 

337°)        1 

336°)        1 

335°)        ] 

334°) 

Lat. 

Dep. 

Lat. 

Dep. 
0.4 

Lat. 

Dep. 

Lat. 

Dep. 
0.4 

Lat. 

Dep. 

Lat. 

Dep. 

1 

0.9 

0.4 

0.9 

0.9 

0.4 

0.9 

0.9 

0.4 

0.9 

0.4 

2 

1.9 

0.7 

1.9 

0.7 

1.8 

0.8 

1.8 

0.8 

1.8 

0.8 

1.8 

0.9 

3 

2.8 

1.1 

2.8 

1.1 

2.8 

1.2 

2.7 

1.2 

2.7 

1.3 

2.7 

1.3 

4 

3.7 

1.4 

3.7 

1.5 

3.7 

1.6 

3.7 

1.6 

3.6 

1.7 

3.6 

1.8 

5 

4.7 

1.8 

4.6 

1.9 

4.6 

2.0 

4.6 

2.0 

4.5 

2.1 

4.5 

2.2 

6 

5.6 

2.2 

5.6 

2.2 

5.5 

2.3 

5.5 

2.4 

5.4 

2.5 

5.4 

2.6 

7 

6.5 

2.5 

6.5 

2.6 

6.4 

2.7 

6.4 

2.8 

6.3 

3.0 

6.3 

3.1 

8 

7.5 

2.9 

7.4 

3.0 

7.4 

3.1 

7.3 

3.3 

7.3 

3.4 

7.2 

3.5 

9 

8.4 

3.2 

8.3 

3.4 

8.3 

3.5 

8.2 

3.7 

8.2 

3.8 

8.1 

3.9 

10 

9.3 

3.6 

9.3 

3.7 

9.2 

3.9 

9.1 

4.1 

9.1 

4.2 

9.0 

4.4 

11 

10.3 

3.9 

10.2 

4.1 

10.1 

4.3 

10.0 

4.5 

10.0 

4.6 

9.9 

4.8 

12 

11.2 

4.3 

11.1 

4.5 

11.0 

4.7 

11.0 

4.9 

10.9 

5.1 

10.8 

5.3 

13 

12.1 

4.7 

12.1 

4.9 

12.0 

5.1 

11.9 

5.3 

11.8 

5.5 

11.7 

5.7 

14 

13.1 

5.0 

13.0 

5.2 

12.9 

5.5 

12.8 

5.7 

12.7 

5.9 

12.6 

6.1 

15 

14.0 

5.4 

13.9 

5.0 

13.8 

5.9 

13.7 

6.1 

13.6 

6.3 

13.5 

6.6 

16 

14.9 

5.7 

14.8 

0.0 

14.7 

6.3 

14.6 

6.5 

14.5 

6.8 

14.4 

7.0 

17 

15.9 

6.1 

15.8 

0.4 

15.6 

6.6 

15.5 

6.9 

15.4 

7.2 

15.3 

7.5 

18 

16.8 

6.5 

16.7 

6.7 

16.6 

7.0 

16.4 

7.3 

16.3 

7.6 

16.2 

7.9 

19 

17.7 

6.8 

17.6 

7.1 

17.5 

7.4 

17.4 

7.7 

17.2 

8.0 

17.1 

8.3 

20 

18.7 

7.2 

18.5 

7.5 

18.4 

7.8 

18.3 

8.1 

18.1 

8.5 

18.0 

8.8 

21 

19.6 

7.5 

19.5 

7.9 

19.3 

8.2 

19.2 

8.5 

19.0 

8.9 

18.9 

9.2 

22 

20.5 

7.9 

20.4 

8.2 

20.3 

8.6 

20.1 

8.9 

19.9 

9.3 

19.8 

9.6 

23 

21.5 

8.2 

21.3 

8.6 

21.2 

9.0 

21.0 

9.4 

20.8 

9.7 

20.7 

10.1 

24 

22.4 

8.6 

22.3 

9.0 

22.1 

9.4 

21.9 

9.8 

21.8 

10.1 

21.6 

10.5 

25 

23.3 

9.0 

23.2 

9.4 

23.0 

9.8 

22.8 

10.2 

22.7 

10.6 

22.5 

11.0 

26 

24.3 

9.3 

24.1 

9.7 

23.9 

10.2 

23.8 

10.6 

23.6 

11.0 

23.4 

11.4 

27 

25.2 

9.7 

25.0 

10.1 

24.9 

10.5 

24.7 

11.0 

24.5 

11.4 

24.3 

11.8 

28 

26.1 

10.0 

26.0 

10.5 

25.8 

10.9 

25.6 

11.4 

25.4 

11.8 

25.2 

12.3 

29 

27.1 

10.4 

26.9 

10.9 

26.7 

11.3 

26.5 

11.8 

26.3 

12.3 

26.1 

12.7 

30 

28.0 

10.8 

27.8 

11.2 

27.6 

11.7 

27.4 

12.2 

27.2 

12.7 

27.0 

13.2 

31 

28.9 

11.1 

28.7 

11.6 

28.5 

12.1 

28.3 

12.6 

28.1 

13.1 

27.9 

13.6 

32 

29.9 

11.5 

29.7 

12.0 

29.5 

12.5 

29.2 

13.0 

29.0 

13.5 

28.8 

14.0 

33 

30.8 

11.8 

30.6 

12.4 

30.4 

12.9 

30.1 

13.4 

29.9 

13.9 

29.7 

14.5 

34 

31.7 

12.2 

31.5 

12.7 

31.3 

13.3 

31.1 

13.8 

30.8 

14.4 

30.6 

14.9 

35 

32.7 

12.5 

32.5 

13.1 

32.2 

13.7 

32.0 

14.2 

31.7 

14.8 

31.5 

15.3 

36 

33.6 

12.9 

33.4 

13.5 

33.1 

14.1 

32.9 

14.0 

32.6 

15.2 

32.4 

15.8 

37 

34.5 

13.3 

34.3 

13.9 

34.1 

14.5 

33.8 

15.0 

33.5 

15.6 

33.3 

16.2 

38 

35.5 

13.6 

35.2 

14.2 

35.0 

14.8 

34.7 

15.5 

34.4 

16.1 

34.2 

16.7 

39 

36.4 

14.0 

36.2 

14.6 

35.9 

15.2 

35.6 

15.9 

35.3 

16.5 

35.1 

17.1 

40 

37.3 

14.3 

37.1 

15.0 

36.8 

15.6 

36.5 

16.3 

36.3 

16.9 

36.0 

17.5 

41 

38.3 

14.7 

38.0 

15.4 

37.7 

16.0 

37.5 

16.7 

37.2 

17.3 

36.9 

18.0 

42 

39.2 

15.1 

38.9 

15.7 

38.7 

16.4 

38.4 

17.1 

38.1 

17.7 

37.7 

18.4 

43 

40.1 

15.4 

39.9 

16.1 

39.6 

16.8 

39.3 

17.5 

39.0 

18.2 

38.6 

18.8 

44 

41.1 

15.8 

40.8 

16. 5 

40.5 

17.2 

40.2 

17.9 

39.9 

18.6 

39.5 

19.3 

45 

42.0 

16.1 

41.7 

16.9 

41.4 

17.6 

41.1 

18.3 

40.8 

19.0 

40.4 

19.7 

46 

42.9 

16.5 

42.7 

17.2 

42:3 

18.0 

42.0 

18.7 

41.7 

19.4 

41.3 

20.2 

47 

43.9 

16.8 

43.6 

17.6 

43.3 

18.4 

42.9 

19.1 

42.6 

19.9 

42.2 

20.6 

48 

44.8 

17.2 

44.5 

18.0 

44.2 

18.8 

43.9 

19.5 

43.5 

20.3 

43.1 

21.0 

49 

45.7 

17.6 

45.4 

18.4 

45.1 

19.1 

44.8 

19.9 

44.4 

20.7 

44.0 

21.5 

50 

46.7 

17.9 

46.4 

18.7 

46.0 

19.5 

45.7 

20.3 

45.3 

21.1 

44.9 

21.9 

100 

93.4 

35.8 

92.7 

37.5 

92.1 

39.1 

91.4 

40.7 

90.6 

42.3 

89.9 

43.8 

200 

186.7 

71.7 

185.4 

74.9 

184.1 

78.1 

182.7 

81.3 

181.3 

84.5 

179.8 

87.7 

300 

280.1 

107.5 

278.2 

112.4 

276.2 

117.2 

274.1 

122.0 

271.9 

126.8 

269.6 

131.5 

400 

373.4 

143.4 

370.9 

149.8 

368.2 

156.3 

365.4 

162.7 

362.5 

169.0 

359.5 

175.4 

500 

466.8 

179.2 

463.6 

187.3 

460.2 

195.4 

456.8 

203.4 

453.1 

211.3 

449.4 

219.2 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

(111°, 

249°, 

(112°, 

248,° 

(113°. 

247°, 

(114° 

246°, 

(115°, 

245°, 

(116°,  244°, 

29 

1°) 

29 

2°) 

29 

3°) 

2S 

4°) 

29, 

5°) 

296°) 

6< 

r 

6  Pt 

.68° 

6' 

7° 

6( 

5° 

51  Pt 

.  65° 

64° 

The  2-Pt.  or  23°  Courses  are :   N.N.E.,  N.N.W.,  S.S.E.,  S.S.W. 


Table  1.    Traverse  Table 


161 


21° 

22° 

2  Pt 

.23° 

24° 

2iPt.  25° 

26° 

(159°,  201°, 

(158°,  202°, 

(157°, 

203°, 

(156°,  204°, 

(155°,  205°, 

(154°,  206°, 

DiST. 

339°) 

338°) 

337°) 

336°) 

335°) 

334°) 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

51 

47.6 

18.3 

47.3 

19.1 

46.9 

19.9 

46.6 

20.7 

46.2 

21.6 

45.8 

22.4 

52 

48.5 

18.6 

48.2 

19.5 

47.9 

20.3 

47.5 

21.2 

47.1 

22.0 

46.7 

22.8 

53 

49.5 

19.0 

49.1 

19.9 

48.8 

20.7 

48.4 

21.6 

48.0 

22.4 

47.6 

23.2 

54 

50.4 

19.4 

50.1 

20.2 

49.7 

21.1 

49.3 

22.0 

48.9 

22.8 

48.5 

23.7 

55 

51.3 

19.7 

51.0 

20.6 

50.6 

21.5 

50.2 

22.4 

49.8 

23.2 

49.4 

24.1 

56 

52.3 

20.1 

51.9 

21.0 

51.5 

21.9 

51.2 

22.8 

50.8 

23.7 

50.3 

24.5 

57 

53.2 

20.4 

52.8 

21.4 

52.5 

22.3 

52.1 

23.2 

51.7 

24.1 

51.2 

25.0 

58 

54.1 

20.8 

53.8 

21.7 

53.4 

22.7 

53.0 

23.6 

52.6 

24.5 

52.1 

25.4 

59 

55.1 

21.1 

54.7 

22.1 

54.3 

23.1 

53.9 

24.0 

53.5 

24.9 

53.0 

25.9 

60 

56.0 

21.5 

55.6 

22.5 

55.2 

23.4 

54.8 

24.4 

54.4 

25.4 

53.9 

26.3 

61 

56.9 

21.9 

56.6 

22.9 

56.2 

23.8 

55.7 

24.8 

55.3 

25.8 

54.8 

26.7 

62 

57.9 

22.2 

57.5 

23.2 

57.1 

24.2 

56.6 

25.2 

56.2 

26.2 

55.7 

27.2 

63 

58.8 

22.6 

58.4 

23.6 

58.0 

24.6 

57.6 

25.6 

57.1 

26.6 

56.6 

27.6 

64 

59.7 

22.9 

59.3 

24.0 

58.9 

25.0 

58.5 

26.0 

58.0 

27.0 

57.5 

28.1 

65 

60.7 

23.3 

60.3 

24.3 

59.8 

25.4 

59.4 

26.4 

58.9 

27.5 

58.4 

28.5 

66 

61.6 

23.7 

61.2 

24.7 

60.8 

25.8 

60.3 

26.8 

59.8 

27.9 

59.3 

28.9 

67 

62.5 

24.0 

62.1 

25.1 

61.7 

26.2 

61.2 

27.3 

60.7 

28.3 

60.2 

29.4 

68 

63.5 

24.4 

63.0 

25.5 

62.6 

26.6 

62.1 

27.7 

61.6 

28.7 

61.1 

29.8 

69 

64.4 

24.7 

64.0 

25.8 

63.5 

27.0 

63.0 

28.1 

62.5 

29.2 

62.0 

30.2 

70 

65.4 

25.1 

64.9 

26.2 

64.4 

27.4 

63.9 

28.5 

63.4 

29.6 

62.9 

30.7 

71 

66.3 

25.4 

65.8 

26.6 

65.4 

27.7 

64.9 

28.9 

64.3 

30.0 

63.8 

31.1 

72 

67.2 

25.8 

66.8 

27.0 

66.3 

28.1 

65.8 

29.3 

65.3 

30.4 

64.7 

31.6 

73 

68.2 

26.2 

67.7 

27.3 

67.2 

28.5 

66.7 

29.7 

66.2 

30.9 

65.6 

32.0 

74 

69.1 

26.5 

68.6 

27.7 

68.1 

28.9 

67.6 

30.1 

67.1 

31.3 

66.5 

32.4 

75 

70.0 

26.9 

69.5 

28.1 

69.0 

29.3 

68.5 

30.5 

68.0 

31.7 

67.4 

32.9 

76 

71.0 

27.2 

70.5 

28.5 

70.0 

29.7 

69.4 

30.9 

68.9 

32.1 

68.3 

33.3 

77 

71.9 

27.6 

71.4 

28.8 

70.9 

30.1 

70.3 

31.3 

69.8 

32.5 

69.2 

33.8 

78 

72.8 

28.0 

72.3 

29.2 

71.8 

30.5 

71.3 

31.7 

70.7 

33.0 

70.1 

34.2 

79 

73.0 

28.3 

73.2 

29.0 

72.7 

30.9 

72.2 

32.1 

71.6 

33.4 

71.0 

34.6 

80 

74.7 

28.7 

74.2 

30.0 

73.6 

31.3 

73.1 

32.5 

72.5 

33.8 

71.9 

35.1 

81 

75.6 

29.0 

75.1 

30.3 

74.6 

31.6 

74.0 

32.9 

73.4 

34.2 

72.8 

35.5 

82 

76.6 

29.4 

76.0 

30.7 

75.5 

32.0 

74.9 

33.4 

74.3 

34.7 

73.7 

35.9 

83 

77.5 

29.7 

77.0 

31.1 

76.4 

32.4 

75.8 

33.8 

75.2 

35.1 

74.6 

36.4 

84 

78.4 

30.1 

77.9 

31.5 

77.3 

32.8 

76.7 

34.2 

76.1 

35.5 

75.5 

36.8 

85 

79.4 

30.5 

78.8 

31.8 

78.2 

33.2 

77.7 

34.6 

77.0 

35.9 

76.4 

37.3 

86 

80.3 

30.8 

79.7 

32.2 

79.2 

33.6 

78.6 

35.0 

77.9 

36.3 

77.3 

37.7 

87 

81.2 

31.2 

80.7 

32.6 

80.1 

34.0 

79.5 

35.4 

78.8 

36.8 

78.2 

38.1 

88 

82.2 

31.5 

81.0 

33.0 

81.0 

34.4 

80.4 

35.8 

79.8 

37.2 

79.1 

38.6 

89 

83.1 

31.9 

82.5 

33.3 

81.9 

34.8 

81.3 

36.2 

80.7 

37.6 

80.0 

39.0 

90 

84.0 

32.3 

83.4 

33.7 

82.8 

35.2 

82.2 

36.6 

81.6 

38.0 

80.9 

39.5 

91 

85.0 

32.6 

84.4 

34.1 

83.8 

35.6 

83.1 

37.0 

82,5 

38.5 

81.8 

39.9 

92 

85.9 

33.0 

85.3 

34.5 

84.7 

35.9 

84.0 

37.4 

83.4 

38.9 

82.7 

40.3 

93 

86.8 

33.3 

86.2 

34.8 

85.6 

36.3 

85.0 

37.8 

84.3 

39.3 

83.6 

40.8 

94 

87.8 

33.7 

87.2 

35.2 

86.5 

36.7 

85.© 

38.2 

85.2 

39.7 

84.5 

41.2 

95 

88:7 

34.0 

88.1 

35.6 

87.4 

37.1 

86.8 

38.6 

86.1 

40.1 

85.4 

41.6 

96 

89.6 

34.4 

89.0 

36.0 

88.4 

37.5 

87.7 

39.0 

87.0 

40.6 

86.3 

42.1 

97 

90.6 

34.8 

89.9 

36.3 

89.3 

37.9 

88.6 

39.5 

87.9 

41.0 

87.2 

42.5 

98 

91.5 

35.1 

90.9 

36.7 

90.2 

38.3 

89.5 

39.9 

88.8 

41.4 

88.1 

43.0 

99 

92.4 

35.5 

91.8 

37.1 

91.1 

38.7 

90.4 

40.3 

89.7 

41.8 

89.0 

43.4 

100 

93.4 

35.8 

92.7 

37.5 

92.1 

39.1 

91.4 

40.7 

90.6 

42.3 

89.9 

43.8 

600 

560.1 

215.0 

556.3 

224.8 

552.3 

234.4 

548.1 

244.0 

543.8 

253.6 

539.3 

263.0 

700 

653.6 

250.8 

649.1 

262.2 

644.3 

273.5 

639.5 

284.7 

634.5 

295.8 

629.2 

306.8 

800 

746.9 

286.7 

741.8 

299.7 

736.4 

312.6 

730.8 

325.4 

725.1 

338.1 

719.1 

350.6 

900 

840.3 

322.5 

834.5 

337.1 

828.3 

351.7 

822.1 

366.0 

815.6 

380.3 

808.9 

394.5 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

(111°,  249°, 

(112°,  248°, 

(113° 

247°, 

(114°,  246°, 

(115°,  245°, 

(116°,  244°, 

291°) 

292°) 

29 

3°) 

294°) 

295°) 

296°) 

69° 

6  Pt.  68° 

6 

7° 

66° 

5f  Pt.65° 

64° 

The  6-Pt.  or  68°  Courses  are:  E.N.E.,  W.N.W.,  E.S.E.,  W.S.W. 


162 


Table  1.    Trayerse  Table 


27° 

2h  Pt 

.28° 

29° 

30°        1 

2f  Pt 

.  31° 

32° 

(153°, 

207°. 

(152°. 

208°, 

(151°. 

209°, 

(150°. 

210°, 

(149°. 

211°, 

(148°,  212°, 

DiST. 

333°) 

332°) 

331°) 

330°) 

329°) 

328°) 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

1 

0.9 

0.5 

0.9 

0.5 

0.9 

0.5 

0.9 

0.5 

0.9 

0.5 

0.8 

0.5 

2 

1.8 

0.9 

1.8 

0.9 

1.7 

1.0 

1.7 

1.0 

1.7 

1.0 

1.7 

1.1 

3 

2.7 

1.4 

2.6 

1.4 

2.6 

1.5 

2.6 

1.5 

2.6 

1.5 

2.5 

1.6 

4 

3.6 

1.8 

3.5 

1.9 

3.5 

1.9 

3.5 

2.0 

3.4 

2.1 

3.4 

2.1 

5 

4.5 

2.3 

4.4 

2.3 

4.4 

2.4 

4.3 

2.5 

4.3 

2.6 

4.2 

2.6 

6 

5.3 

2.7 

5.3 

2.8 

5.2 

2.9 

5.2 

3.0 

5.1 

3.1 

5.1 

3.2 

7 

6.2 

3.2 

6.2 

3.3 

6.1 

3.4 

6.1 

3.5 

6.0 

3.6 

5.9 

3.7 

8 

7.1 

3.6 

7.1 

3.8 

7.0 

3.9 

6.9 

4.0 

6.9 

4.1 

6.8 

4.2 

9 

8.0 

4.1 

7.9 

4.2 

7.9 

4.4 

7.8 

4.5 

7.7 

4.6 

7.6 

4.8 

10 

8.9 

4.5 

8.8 

4.7 

8.7 

4.8 

8.7 

5.0 

8.6 

5.2 

8.5 

5.3 

11 

9.8 

5.0 

9.7 

5.2 

9.6 

5.3 

9.5 

5.5 

9.4 

5.7 

9.3 

5.8 

12 

10.7 

5.4 

10.6 

5.6 

10.5 

5.8 

10.4 

6.0 

10.3 

6.2 

10.2 

6.4 

13 

11.6 

5.9 

11.5 

6.1 

11.4 

6.3 

11.3 

6.5 

11.1 

6.7 

11.0 

6.9 

14 

12.5 

6.4 

12.4 

6.6 

12.2 

6.8 

12.1 

7.0 

12.0 

7.2 

11.9 

7.4 

15 

13.4 

6.8 

13.2 

7.0 

13.1 

7.3 

13.0 

7.5 

12.9 

7.7 

12.7 

7.9 

16 

14.3 

7.3 

14.1 

7.5 

14.0 

7.8 

13.9 

8.0 

13.7 

8.2 

13.6 

8.5 

17 

15.1 

7.7 

15.0 

8.0 

14.9 

8.2 

14.7 

8.5 

14.6 

8.8 

14.4 

9.0 

18 

16.0 

8.2 

15.9 

8.5 

15.7 

8.7 

15.6 

9.0 

15.4 

9.3 

15.3 

9.5 

19 

16.9 

8.6 

16.8 

8.9 

16.6 

9.2 

16.5 

9.5 

16.3 

9.8 

16.1 

10.1 

20 

17.8 

9.1 

17.7 

9.4 

17.5 

9.7 

17.3 

10.0 

17.1 

10.3 

17.0 

10.6 

21 

18.7 

9.5 

18.5 

9.9 

18.4 

10.2 

18.2 

10.5 

18.0 

10.8 

17.8 

11.1 

22 

19.6 

10.0 

19.4 

10.3 

19.2 

10.7 

19.1 

11.0 

18.9 

11.3 

18.7 

11.7 

23 

20.5 

10.4 

20.3 

10.8 

20.1 

11.2 

19.9 

11.5 

19.7 

11.8 

19.5 

12.2 

24 

21.4 

10.9 

21.2 

11.3 

21.0 

11.6 

20.8 

12.0 

20.6 

12.4 

20.4 

12.7 

25 

22.3 

11.3 

22.1 

11.7 

21.9 

12.1 

21.7 

12.5 

21.4 

12.9 

21.2 

13.2 

26 

23.2 

11.8 

23.0 

12.2 

22.7 

12.6 

22.5 

13.0 

22.3 

13.4 

22.0 

13.8 

27 

24.1 

12.3 

23.8 

12.7 

23.6 

13.1 

23.4 

13.5 

23.1 

13.9 

22.9 

14.3 

28 

24.9 

12.7 

24.7 

13.1 

24.5 

13.6 

24.2 

14.0 

24.0 

14.4 

23.7 

14.8 

29 

25.8 

13.2 

25.6 

13.6 

25.4 

14.1 

25.1 

14.5 

24.9 

14.9 

24.6 

15.4 

30 

26.7 

13.6 

26.5 

14.1 

26.2 

14.5 

26.0 

15.0 

25.7 

15.5 

25.4 

15.9 

31 

27.6 

14.1 

27.4 

14.6 

27.1 

15.0 

26.8 

15.5 

26.6 

16.0 

26.3 

16.4 

32 

28.5 

14.5 

28.3 

15.0 

28.0 

15.5 

27.7 

16.0 

27.4 

16.5 

27.1 

17.0 

33 

29.4 

15.0 

29.1 

15.5 

28.9 

16.0 

28.6 

16.5 

28.3 

17.0 

28.0 

17.5 

34 

30.3 

15.4 

30.0 

16.0 

29.7 

16.5 

29.4 

17.0 

29.1 

17.5 

28.8 

18.0 

35 

31.2 

15.9 

30.9 

16.4 

30.6 

17.0 

30.3 

17.5 

30.0 

18.0 

29.7 

18.5 

36 

32.1 

16.3 

31.8 

16.9 

31.5 

17.5 

31.2 

18.0 

30.9 

18.5 

30.5 

19.1 

37 

33.0 

16.8 

32.7 

17.4 

32.4 

17.9 

32.0 

18.5 

31.7 

19.1 

31.4 

19.6 

38 

33.9 

17.3 

33.6 

17.8 

33.2 

18.4 

32.9 

19.0 

32.6 

19.6 

32.2 

20.1 

39 

34.7 

17.7 

34.4 

18.3 

34.1 

18.9 

33.8 

19.5 

33.4 

20.1 

33.1 

20.7 

40 

35.6 

18.2 

35.3 

18.8 

35.0 

19.4 

34.6 

20.0 

34.3 

20.6 

33.9 

21.2 

41 

36.5 

18.6 

36.2 

19.2 

35.9 

19.9 

35.5 

20.5 

35.1 

21.1 

34.8 

21.7 

42 

37.4 

19.1 

37.1 

19.7 

36.7 

20.4 

36.4 

21.0 

36.0 

21.6 

35.6 

22.3 

43 

38.3 

19.5 

38.0 

20.2 

37.6 

20.8 

37.2 

21.5 

36.9 

22.1 

36.5 

22.8 

44 

39.2 

20.0 

38.8 

20.7 

38.5 

21.3 

38.1 

22.0 

37.7 

22.7 

37.3 

23.3 

45 

40.1 

20.4 

39.7 

21.1 

39.4 

21.8 

39.0 

22.5 

38.6 

23.2 

38.2 

23.8 

46 

41.0 

20.9 

40.6 

21.6 

40.2 

22.3 

39.8 

23.0 

39.4 

23.7 

39.0 

24.4 

47 

41.9 

21.3 

41.5 

22.1 

41.1 

22.8 

40.7 

23.5 

40.3 

24.2 

39.9 

24.9 

48 

42.8 

21.8 

42.4 

22.5 

42.0 

23.3 

41.6 

24.0 

41.1 

24.7 

40.7 

25.4 

49 

43.7 

22.2 

43.3 

23.0 

42.9 

23.8 

42.4 

24.5 

42.0 

25.2 

41.6 

26.0 

50 

44.6 

22.7 

44.1 

23.5 

43.7 

24.2 

43.3 

25.0 

42.9 

25.8 

42.4 

26.5 

100 

89.1 

45.4 

88.3 

46.9 

87.5 

48.5 

86.6 

50.0 

85.7 

51.6 

84.8 

53.0 

200 

178.2 

90.8 

176.6 

93.9 

174.9 

97.0 

173.2 

100.0 

171.4 

103.0 

169.6 

106.0 

300 

267.3 

136.2 

264.9 

140.8 

262.4 

145.4 

259.8 

150.0 

257.1 

154.5 

254.4 

159.0 

400 

.356.4 

181.6 

353.1 

187.8 

.349.8 

193.9 

346.4 

200.0 

342.9 

206.0 

339.2 

211.9 

500 

445.5 

227.0 

441.5 

234.7 

437.3 

242.4 

433.0 

250.0 

428.6 

257.5 

424.0 

265.0 

Dep. 

Lat. 

Dep. 

1  Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

(121°. 

Lat. 
239°, 

Dep. 

Lat. 

(117°, 

243°, 

(118° 

242°, 

(119° 

241°, 

(120° 

240°, 

(122°.  238°, 

29 

7°) 

29 

8°) 

29 

9°) 

30 

0°) 

30 

1°) 

302°) 

6: 

J° 

5§P1 

t.  62° 

€ 

.1° 

6 

0° 

5iPt 

.  59° 

58° 

Table  1.    Traverse  Table 


163 


27° 

2i  Pt.  28° 

29° 

30° 

2f  Pt.  31° 

32° 

(153°,  207°, 

(152°,  208°, 

(151°,  209°, 

(150°,  210°, 

(149°,  211°, 

(148°,  212°, 

DiST. 

333°) 

332°) 

331°) 

330°) 

329°) 

328°) 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

51 

45.4 

23.2 

45.0 

23.9 

44.6 

24.7 

44.2 

25.5 

43.7 

26.3 

43.3 

27.0 

52 

46.3 

23.6 

45.9 

24.4 

45.5 

25.2 

45.0 

26.0 

44.6 

26.8 

44.1 

27.6 

53 

47.2 

24.1 

46.8 

24.9 

46.4 

25.7 

45.9 

26.5 

45.4 

27.3 

44.9 

28.1 

54 

48.1 

24.5 

47.7 

25.4 

47.2 

26.2 

46.8 

27.0 

46.3 

27.8 

45.8 

28.6 

55 

49.0 

25.0 

48.6 

25.8 

48.1 

26.7 

47.6 

27.5 

47.1 

28.3 

46.6 

29.1 

56 

49.9 

25.4 

49.4 

26.3 

49.0 

27.1 

48.5 

28.0 

48.0 

28.8 

47.5 

29.7 

57 

50.8 

25.9 

50.3 

26.8 

49.9 

27.6 

49.4 

28.5 

48.9 

29.4 

48.3 

30.2 

58 

51.7 

26.3 

51.2 

27.2 

50.7 

28.1 

50.2 

29.0 

49.7 

29.9 

49.2 

30.7 

59 

52.6 

26.8 

52.1 

27.7 

51.6 

28.6 

51.1 

29.5 

50.6 

30.4 

50.0 

31.3 

60 

53.5 

27.2 

53.0 

28.2 

52.5 

29.1 

52.0 

30.0 

51.4 

30.9 

50.9 

31.8 

61 

54.4 

27.7 

53.9 

28.6 

53.4 

29.6 

52.8 

30.5 

52.3 

31.4 

51.7 

32.3 

62 

55.2 

28.1 

54.7 

29.1 

54.2 

30.1 

53.7 

31.0 

53.1 

31.9 

52.6 

32.9 

63 

56.1 

28.6 

55.6 

29.6 

55.1 

30.5 

54.6 

31.5 

54.0 

32.4 

53.4 

33.4 

64 

57.0 

29.1 

56.5 

30.0 

56.0 

31.0 

55.4 

32.0 

54.9 

33.0 

54.3 

33.9 

65 

57.9 

29.5 

57.4 

30.5 

56.9 

31.5 

56.3 

32.5 

55.7 

33.5 

55.1 

34.4 

66 

58.8 

30.0 

58.3 

31.0 

57.7 

32.0 

57.2 

33.0 

56.6 

34.0 

56.0 

35.0 

67 

59.7 

30.4 

59.2 

31.5 

58.6 

32.5 

58.0 

33.5 

57.4 

34.5 

56.8 

35.5 

68 

60.6 

30.9 

60.0 

31.9 

59.5 

33.0 

58.9 

34.0 

58.3 

35.0 

57.7 

36.0 

69 

61.5 

31.3 

60.9 

32.4 

60.3 

33.5 

59.8 

34.5 

59.1 

35.5 

58.5 

36.6 

70 

62.4 

31.8 

61.8 

32.9 

61.2 

33.9 

60.6 

35.0 

60.0 

36.1 

59.4 

37.1 

71 

63.3 

32.2 

62.7 

33.3 

62.1 

34.4 

61.5 

35.5 

60.9 

36.6 

60.2 

37.6 

72 

64.2 

32.7 

63.6 

33.8 

63.0 

34.9 

62.4 

36.0 

61.7 

37.1 

61.1 

38.2 

73 

65.0 

33.1 

64.5 

34.3 

63.8 

35.4 

63.2 

36.5 

62.6 

37.6 

61.9 

38.7 

74 

65.9 

33.6 

65.3 

34.7 

64.7 

35.9 

64.1 

37.0 

63.4 

38.1 

62.8 

39.2 

75 

66.8 

34.0 

66.2 

35.2 

65.6 

36.4 

65.0 

37.5 

64.3 

38.6 

63.6 

39.7 

76 

67.7 

34.5 

67.1 

35.7 

66.5 

36.8 

65.8 

38.0 

65.1 

39.1 

64.5 

40.3 

77 

68.6 

35.0 

68.0 

36.1 

67.3 

37.3 

66.7 

38.5 

66.0 

39.7 

65.3 

40.8 

78 

69.5 

35.4 

68.9 

36.6 

68.2 

37.8 

67.5 

39.0 

66.9 

40.2 

66.1 

41.3 

79 

70.4 

35.9 

69.8 

37.1 

69.1 

38.3 

68.4 

39.5 

67.7 

40.7 

67.0 

41.9 

80 

71.3 

36.3 

70.6 

37.6 

70.0 

38.8 

69.3 

40.0 

68.6 

41.2 

67.8 

42.4 

81 

72.2 

36.8 

71.5 

38.0 

70.8 

39.3 

70.1 

40.5 

69.4 

41.7 

68.7 

42.9 

82 

73.1 

37.2 

72.4 

38.5 

71.7 

39.8 

71.0 

41.0 

70.3 

42.2 

69.5 

43.5 

83 

74.0 

37.7 

73.3 

39.0 

72.6 

40.2 

71.9 

41.5 

71.1 

42.7 

70.4 

44.0 

84 

74.8 

38.1 

74.2 

39.4 

73.5 

40.7 

72.7 

42.0 

72.0 

43.3 

71.2 

44.5 

85 

75.7 

38.6 

75.1 

39.9 

74.3 

41.2 

73.6 

42.5 

72.9 

43.8 

72.1 

45.0 

86 

76.6 

39.0 

75.9 

40.4 

75.2 

41.7 

74.5 

43.0 

73.7 

44.3 

72.9 

45.6 

87 

77.5 

39.5 

76.8 

40.8 

-76.1 

42.2 

75.3 

43.5 

74.6 

44.8 

73.8 

46.1 

88 

78.4 

40.0 

77.7 

41.3 

77.0 

42.7 

76.2 

44.0 

75.4 

45.3 

74.6 

46.6 

89 

79.3 

40.4 

78.6 

41.8 

77.8 

43.1 

77.1 

44.5 

76.3 

45.8 

75.5 

47.2 

90 

80.2 

40.9 

79.5 

42.3 

78.7 

43.6 

77.9 

45.0 

77.1 

46.4 

76.3 

47.7 

91 

81.1 

41.3 

80.3 

42.7 

79.6 

44.1 

78.8 

45.5 

78.0 

46.9 

77.2 

48.2 

92 

82.0 

41.8 

81.2 

43.2 

80.5 

44.6 

79.7 

46.0 

78.9 

47.4 

78.0 

48.8 

93 

82.9 

42.2 

82.1 

43.7 

81.3 

45.1 

80.5 

46.5 

79.7 

47.9 

78.9 

49.3 

94 

83.8 

42.7 

83.0 

44.1 

82.2 

45.6 

81.4 

47.0 

80.6 

48.4 

79.7 

49.8 

95 

84.6 

43.1 

83.9 

44.6 

83.1 

46.1 

82.3 

47.5 

81.4 

48.9 

80.6 

50.3 

96 

85.5 

43.6 

84.8 

45.1 

84.0 

46.5 

83.1 

48.0 

82.3 

49.4 

81.4 

50.9 

97 

86.4 

44.0 

85.6 

45.5 

84.8 

47.0 

84.0 

48.5 

83.1 

50.0 

82.3 

51.4 

98 

87.3 

44.5 

86.5 

46.0 

85.7 

47.5 

84.9 

49.0 

84.0 

50.5 

83.1 

51.9 

99 

88.2 

44.9 

87.4 

46.5 

86.6 

48.0 

85.7 

49.5 

84.9 

51.0 

84.0 

52.5 

100 

89.1 

45.4 

88.3 

46.9 

87.5 

48.5 

86.6 

50.0 

85.7 

51.5 

84.8 

53,0 

600 

534.6 

272.4 

529.8 

281.7 

524.8 

290.9 

519.6 

300.0 

514.3 

309.0 

508.8 

318.0 

700 

623.7 

317.8 

618.0 

328.6 

612.2 

339.4 

606.1 

350.0 

600.1 

360.4 

593.6 

371.0 

800 

712.9 

363.2 

706.3 

375.6 

699.7 

387.9 

692.8 

400.0 

685.8 

412.0 

678.4 

423.9 

900 

801.9 

408.5 

794.5 

422.5 

787.0 

436.3 

779.3 

450.0 

771.4 

463.4 

763.2 

476.8 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

(117°,  243°, 

(118°,  242°, 

(119°,  241°, 

(120°,  240° 

(121°,  239°, 

(122°,  238°, 

297°) 

298°) 

299°) 

300°) 

301°) 

302°) 

63° 

5§  Pt.  62° 

61° 

60° 

51  Pt.  59° 

58° 

164 


Table  1.    Traverse  Table 


33° 

3  Pt.  34° 

35° 

36° 

3}  Pt.  37° 

38° 

(147°,  213°, 

(146°,  214°, 

(145°,  215°, 

(144°,  216°, 

(143°,  217°, 

(142°,  218°, 

DiST. 

327°) 

326°) 

325°) 

324°) 

323°) 

322°) 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat.  1  Dep. 

Lat. 

Dep. 

1 

0.8 

0.5 

0.8 

0.6 

0.8 

0.6 

0.8 

0.6 

0.8 

0.6 

0.8 

0.6 

2 

1.7 

1.1 

1.7 

1.1 

1.6 

1.1 

1.6 

1.2 

1.6 

1.2 

1.6 

1.2 

3 

2.5 

1.6 

2.5 

1.7 

2.5 

1.7 

2.4 

1.8 

2.4 

1.8 

2.4 

1.8 

4 

3.4 

2.2 

3.3 

2.2 

3.3 

2.3 

3.2 

2.4 

3.2 

2.4 

3.2 

2.5 

5 

4.2 

2.7 

4.1 

2.8 

4.1 

2.9 

4.0 

2.9 

4.0 

3.0 

3.9 

3.1 

G 

5.0 

3.3 

5.0 

3.4 

4.9 

3.4 

4.9 

3.5 

4.8 

3.6 

4.7 

3.7 

7 

5.9 

3.8 

5.8 

3.9 

5.7 

4.0 

5.7 

4.1 

5.6 

4.2 

5.5 

4.3 

8 

6.7 

4.4 

0.6 

4.5 

6.6 

4.6 

6.5 

4.7 

6.4 

4.8 

6.3 

4.9 

9 

7.5 

4.9 

7.5 

5.0 

7.4 

5.2 

7.3 

5.3 

7.2 

5.4 

7.1 

5.5 

10 

8.4 

5.4 

8.3 

5.6 

8.2 

5.7 

8.1 

5.9 

8.0 

6.0 

7.9 

6.2 

11 

9.2 

6.0 

9.1 

6.2 

9.0 

6.3 

8.9 

6.5 

8.8 

6.6 

8.7 

6.8 

12 

10.1 

6.5 

9.9 

6.7 

9.8 

6.9 

^.7 

7.1 

9.6 

7.2 

9.5 

7.4 

13 

10.9 

7.1 

10.8 

7.3 

10.6 

7.5 

10.5 

7.6 

10.4 

7.8 

10.2 

8.0 

14 

11.7 

7.6 

11.6 

7.8 

11.5 

8.0 

11.3 

8.2 

11.2 

8.4 

11.0 

8.6 

15 

12.6 

8.2 

12.4 

8.4 

12.3 

8.6 

12.1 

8.8 

12.0 

9.0 

11.8 

9.2 

IG 

13.4 

8.7 

13.3 

8.9 

13.1 

9.2 

12.9 

9.4 

12.8 

9.6 

12.6 

9.9 

17 

14.3 

9.3 

14.1 

9.5 

13.9 

9.8 

13.8 

10.0 

13.G 

10.2 

13.4 

10.5 

18 

15.1 

9.8 

14.9 

10.1 

14.7 

10.3 

14.6 

10.6 

14.4 

10.8 

14.2 

11.1 

19 

15.9 

10.3 

15.8 

10.6 

15.6 

10.9 

15.4 

11.2 

15.2 

11.4 

15.0 

11.7 

20 

16.8 

10.9 

16.6 

11.2 

16.4 

11.5 

16.2 

11.8 

16.0 

12.0 

15.8 

12.3 

21 

17.6 

11.4 

17.4 

11.7 

17.2 

12.0 

17.0 

12.3 

16.8 

12.6 

16.5 

12.9 

22 

18.5 

12.0 

18.2 

12.3 

18.0 

12.0 

17.8 

12.9 

17.6 

13.2 

17.3 

13.5 

23 

19.3 

12.5 

19.1 

12.9 

18.8 

13.2 

18.6 

13.5 

18.4 

13.8 

18.1 

14.2 

24 

20.1 

13.1 

19.9 

13.4 

19.7 

13.8 

19.4 

14.1 

19.2 

14.4 

18.9 

14.8 

25 

21.0 

13.6 

20.7 

14.0 

20.5 

14.3 

20.2 

14.7 

20.0 

15.0 

19.7 

15.4 

26 

21.8 

14.2 

21.6 

14.5 

21.3 

14.9 

21.0 

15.3 

20.8 

15.6 

20.5 

16.0 

27 

22.6 

14.7 

22.4 

15.1 

22.1 

15.5 

21.8 

15.9 

21.6 

16.2 

21.3 

16.6 

28 

23.5 

15.2 

23.2 

15.7 

22.9 

16.1 

22.7 

16.5 

22.4 

16.9 

22.1 

17.2 

29 

24.3 

15.8 

24.0 

16.2 

23.8 

,16.6 

23.5 

17.0 

23.2 

17.5 

22.9 

17.9 

30 

25.2 

16.3 

24.9 

16.8 

24.6 

17.2 

24.3 

17.6 

24.0 

18.1 

23.6 

18.5 

31 

26.0 

16.9 

25.7 

17.3 

25.4 

17.8 

25.1 

18.2 

24.8 

18.7 

24.4 

19.1 

32 

26.8 

17.4 

26.5 

17.9 

26.2 

18.4 

25.9 

18.8 

25.6 

19.3 

25.2 

19.7 

33 

27.7 

18.0 

27.4 

18.5 

27.0 

18.9 

26.7 

19.4 

26.4 

19.9 

26.0 

20.3 

34 

28.5 

18.5 

28.2 

19.0 

27.9 

19.5 

27.5 

20.0 

27.2 

20.5 

26.8 

20.9 

35 

29.4 

19.1 

29.0 

19.6 

28.7 

20.1 

28.3 

20.6 

28.0 

21.1 

27.6 

21.5 

3G 

30.2 

19.6 

29.8 

20.1 

29.5 

20.6 

29.1 

21.2 

28.8 

21.7 

28.4 

22.2 

37 

31.0 

20.2 

30.7 

20.7 

30.3 

21.2 

29.9 

21.7 

29.5 

22.3 

29.2 

22.8 

38 

31.9 

20.7 

31.5 

21.2 

31.1 

21.8 

30.7 

22.3 

30.3 

22.9 

29.9 

23.4 

39 

32.7 

21.2 

32.3 

21.8 

31.9 

22.4 

31.6 

22.9 

31.1 

23.5 

30.7 

24.0 

40 

33.5 

21.8 

33.2 

22.4 

32.8 

22.9 

32.4 

23.5 

31.9 

24.1 

31.5 

24.6 

41 

34.4 

22.3 

34.0 

22.9 

33.6 

23.5 

33.2 

24.1 

32.7 

24.7 

32.3 

25.2 

42 

35.2 

22.9 

34.8 

23.5 

34.4 

24.1 

34.0 

24.7 

33.5 

25.3 

33.1 

25.9 

43 

36.1 

23.4 

35.6 

24.0 

35.2 

24.7 

34.8 

25.3 

34.3 

25.9 

33.9 

26.5 

44 

36.9 

24.0 

36.5 

24.6 

36.0 

25.2 

35.6 

25.9 

35.1 

26.5 

34.7 

27.1 

45 

37.7 

24.5 

37.3 

25.2 

36.9 

25.8 

36.4 

26.5 

35.9 

27.1 

35.5 

27.7 

4G 

38.6 

25.1 

38.1 

25.7 

37.7 

26.4 

37.2 

27.0 

36.7 

27.7 

36.2 

28.3 

47 

39.4 

25.6 

39.0 

26.3 

38.5 

27.0 

38.0 

27.6 

37.5 

28.3 

37.0 

28.9 

48 

40.3 

26.1 

39.8 

26.8 

39.3 

27.5 

38.8 

28.2 

38.3 

28.9 

37.8 

29.6 

49 

41.1 

26.7 

40.6 

27.4 

40.1 

28.1 

39.6 

28.8 

39.1 

29.5 

38.6 

30.2 

50 

41.9 

27.2 

41.5 

28.0 

41.0 

28.7 

40.5 

29.4 

39.9 

30.1 

39.4 

30.8 

100 

83.9 

54.5 

82.9 

55.9 

81.9 

57.4 

80. C 

58.8 

79.9 

60.2 

78.8 

61.6 

200 

107.7 

108.9 

165.8 

111.8 

163.8 

114.7 

161.8 

117.6 

159.7 

120.4 

157.6 

123.1 

300 

251.6 

163.4 

248.7 

167.8 

245.7 

172.1 

242.7 

176.3 

239.6 

180.5 

236.4 

184.7 

400 

335.5 

217.8 

331.6 

223.7 

.327.7 

229.4 

323.6 

235.1 

319.4 

240.7 

315.2 

246.3 

500 

419.3 

272.3 

414.5 

279.6 

409.6  286.8 

404.5 

293.9 

399.3 

300.9 

394.0 

307.8 

Dep. 

Lat. 

Dep. 

Lat. 

Dep.     Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

(123°,  237°, 

(124°,  236°, 

(125°,  235°, 

(126°,  234°, 

(127°,  233°, 

(128°,  232°, 

303°) 

304°) 

305°) 

306°) 

307°) 

308°) 

57° 

5  Pt.  56° 

55° 

54° 

4f  Pt.  53° 

52° 

The  3-Pt.  or  34°  Courses  are :  N.E.  by  N.,  N.W.  by  N.,  S.E.  by  S.,  S.W.  by  S. 


Table  1.    Traverse  Table 


165 


33° 

3  Pt.  34° 

35° 

36° 

3iPt.37° 

38° 

(147°,  213°, 

J(146°,  214°, 

(145°,  215°, 

(144°,  216°, 

(143° 

217°, 

(142°,  218°, 

DiST. 

327°) 

326°) 

325°) 

324°) 

323°) 

322°) 

Lat. 

Dep. 

[hat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

51 

42.8 

27.8 

42.3 

28.5 

41.8 

29.3 

41.3 

30.0 

40.7 

30.7 

40.2 

31.4 

52 

43.6 

28.3 

43.1 

29.1 

42.6 

29.8 

42.1 

30.G 

41.5 

31.3 

41.0 

32.0 

53 

44.4 

28.9 

43.9 

29.6 

43.4 

30.4 

42.9 

31.2 

42.3 

31.9 

41.8 

32.6 

54 

45.3 

29.4 

44.8 

30.2 

44.2 

31.0 

43.7 

31.7 

43.1 

32.5 

42.6 

33.2 

55 

46.1 

30.0 

45.6 

30.8 

45.1 

31.5 

44.5 

32.3 

43.9 

33.1 

43.3 

33.9 

56 

47.0 

30.5 

46.4 

31.3 

45.9 

32.1 

45.3 

32.9 

44.7 

33.7 

44.1 

34.5 

57 

47.8 

31.0 

47.3 

31.9 

46.7 

32.7 

46.1 

33.5 

45.5 

34.3 

44.9 

35.1 

58 

48.6 

31.6 

48.1 

32.4 

47.5 

33.3 

46.9 

34.1 

46.3 

34.9 

45.7 

35.7 

59 

49.5 

32.1 

48.9 

33.0 

48.3 

33.8 

47.7 

34.7 

47.1 

35.5 

46.5 

36.3 

60 

50.3 

32.7 

49.7 

33.6 

49.1 

34.4 

48.5 

35.3 

47.9 

36.1 

47.3 

36.9 

61 

51.2 

33.2 

50.6 

34.1 

50.0 

35.0 

49.4 

35.9 

48.7 

36.7 

48.1 

37.6 

62 

52.0 

33.8 

51.4 

34.7 

50.8 

35.6 

50.2 

36.4 

49.5 

37.3 

48.9 

38.2 

63 

52.8 

34.3 

52.2 

35.2 

51.6 

36.1 

51.0 

37.0 

50.3 

37.9 

49.6 

38.8 

64 

53.7 

34.9 

53.1 

35.8 

52.4 

36.7 

51.8 

37.6 

51.1 

38.5 

50.4 

39.4 

65 

54.5 

35.4 

53.9 

36.3 

53.2 

37.3 

52.6 

38.2 

51.9 

39.1 

51.2 

40.0 

66 

55.4 

35.9 

54.7 

36.9 

54.1 

37.9 

53.4 

38.8 

52.7 

39.7 

52.0 

40.6 

67 

56.2 

36.5 

55.5 

37.5 

54.9i  38.4 

54.2 

39.4 

53.5 

40.3 

52.8 

41.2 

68 

57.0 

37.0 

56.4 

38.0 

55.7 

39.0 

55.0 

40.0 

54.3 

40.9 

53.6 

41.9 

69 

57.9 

37.6 

57.2 

38.6 

56.5 

39.6 

55.8 

40.6 

55.1 

41.5 

54.4 

42.5 

70 

58.7 

38.1 

58.0 

39.1 

57.3 

40.2 

56.6 

41.1 

55.9 

42.1 

55.2 

43.1 

71 

59.5 

38.7 

58.9 

39.7 

58.2 

40.7 

57.4 

41.7 

56.7 

42.7 

55.9 

43.7 

72 

60.4 

39.2 

59.7 

40.3 

59.0 

41.3 

58.2 

42.3 

57.5 

43.3 

56.7 

44.3 

73 

61.2 

39.8 

60.5 

40.8 

59.8 

41.9 

59.1 

42.9 

58.3 

43.9 

57.5 

44.9 

74 

62.1 

40.3 

61.3 

41.4 

60.6 

42.4 

59.9 

43.5 

59.1 

44.5 

58.3 

45.6 

75 

62.9 

40.8 

62.2 

41.9 

61.4 

43.0 

60.7 

44.1 

59.9 

45.1 

59.1 

46.2 

76 

63.7 

41.4 

63.0 

42.5 

62.3 

43.6 

61.5 

44.7 

60.7 

45.7 

59.9 

46.8 

77 

64.6 

41.9 

63.8 

43.1 

63.1 

44.2 

62.3 

45.3 

61.5 

46.3 

60.7 

47.4 

78 

65.4 

42.5 

64.7 

43.6 

63.9 

44.7 

63.1 

45.8 

62.3 

46.9 

61.5 

48.0 

79 

66.3 

43.0 

65.5 

44.2 

64.7 

45.3 

63.9 

46.4 

63.1 

47.5 

62.3 

48.6 

80 

67.1 

43.6 

66.3 

44.7 

65.5 

45.9 

64.7 

47.0 

63.9 

48.1 

63.0 

49.3 

81 

67.9 

44.1 

67.2 

45.3 

66.4 

46.5 

65.5 

47.6 

64.7 

48.7 

63.8 

49.9 

82 

68.8 

44.7 

68.0 

45.9 

67.2 

47.0 

66.3 

48.2 

65.5 

49.3 

64.6 

50.5 

83 

69.6 

45.2 

68.8 

46.4 

68.0 

47.6 

67.1 

48.8 

66.3 

50.0 

65.4 

51.1 

84 

70.4 

45.7 

69.6 

47.0 

68.8 

48.2 

68.0 

49.4 

67.1 

50.6 

66.2 

51.7 

85 

71.3 

46.3 

70.5 

47.5 

69.6 

48.8 

68.8 

50.0 

67.9 

51.2 

67.0 

52.3 

86 

72.1 

46.8 

71.3 

48.1 

70.4 

49.3 

69.0 

50.5 

68.7 

51.8 

67.8 

52.9 

87 

73.0 

47.4 

72.1 

48.6 

71.3 

49.9 

70.4 

51.1 

69.5 

52.4 

68.6 

53.6 

88 

73.8 

47.9 

73.0 

49.2 

72.1 

50.5 

71.2 

51.7 

70.3 

53.0 

69.3 

54.2 

89 

74.6 

48.5 

73.8 

49.8 

72.9 

51.0 

72.0 

52.3 

71.1 

53.6 

70.1 

54.8 

90 

75.5 

49.0 

74.6 

50.3 

73.7 

51.6 

72.8 

52.9 

71.9 

54.2 

70.9 

55.4 

91 

76.3 

49.6 

75.4 

50.9 

74.5 

52.2 

73.6 

53.5 

72.7 

54.8 

71.7 

56.0 

92 

77.2 

50.1 

76.3 

51.4 

75.4 

52.8 

74.4 

54.1 

73.5 

55.4 

72.5 

56.6 

93 

78.0 

50.7 

77.1 

52.0 

76.2 

53.3 

75.2 

54.7 

74.3 

56.0 

73.3 

57.3 

94 

78.8 

51.2 

77.9 

52.6 

77.0 

53.9 

76.0 

55.3 

75.1' 

56.6 

74.1 

57.9 

95 

79.7 

51.7 

78.8 

53.1 

77.8 

54.5 

76.9 

55.8 

75.9 

57.2 

74.9 

58.5 

96 

80.5 

52.3 

79.6 

53.7 

78.6 

55.1 

77.7 

56.4 

76.7 

57.8 

75.6 

59.1 

97 

81.4 

52.8 

80.4 

54.2 

79.5 

55.6 

78.5 

57.0 

77.5 

58.4 

76.4 

59.7 

98 

82.2 

53.4 

81.2 

54.8 

80.3 

56.2 

79.3 

57.6 

78.3 

59.0 

77.2 

60.3 

99 

83.0 

53.9 

82.1 

55.4 

81.1 

56.8 

80.1 

58.2 

79.1 

59.6 

78.0 

61.0 

100 

83.9 

54.5 

82.9 

55.9 

81.9 

57.4 

80.9 

58.8 

79.9 

60.2 

78.8 

61.6 

600 

503.2 

326.8 

497.4 

335.5 

491.5 

344.1 

485.4 

352.7 

479.2 

361.1 

472.8 

369.4 

700 

587.0 

381.3 

580.3 

391.4 

573.5 

401.5 

566.2 

411.4 

559.0 

421.3 

551.6 

430.8 

800 

671.0 

435.7 

663.3 

447.4 

655.4 

458.8 

647.3 

470.2 

638.9 

481.5 

630.4 

492.5 

900 

754.8 

490.1 

746.1 

503.2 

737.2 

516.2 

728.1 

528.9 

718.6 

541.7 

709.1 

554.0 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

(123°,  237°, 

(124°,  236°, 

(125°,  235°, 

(126°,  234°, 

(127°, 

233°. 

(128°,  232°, 

303°) 

304°) 

305°) 

306°) 

30' 

n 

308°) 

57° 

5  Pt.  56° 

55° 

54° 

4!Pt 

.53° 

52° 

The  5-Pt.  or  56°  Courses  are :  N.E.  by  E.,  S.E.  by  E.,  N.W.  by  W.,  S.W.  by  W. 


166 


Table  1.    Traverse  Table 


3|  Pt.  39° 

40 

41° 

31  Pt.  42° 

43° 

44° 

4  Pt.  45° 

DlST. 

(141° 

219°. 

(140° 

220°, 

(139° 

221°, 

(138°, 

222°, 

(137°,  223°, 

(136°,  224°, 

(135°,  225°, 

321°) 

320°) 

319°) 

318°) 

317°) 

316°) 

315°) 

Lat. 

Dep. 

Lat. 

Dep. 
0.6 

Lat. 
0.8 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 
0.7 

1 

0.8 

0.6 

0.8 

0.7 

0.7 

0.7 

0.7 

0.7 

0.7 

0.7 

0.7 

2 

1.6 

1.3 

1.5 

1.3 

1.5 

1.3 

1.5 

1.3 

1.5 

1.4 

1.4 

1.4 

1.4 

1.4 

3 

2.3 

1.9 

2.3 

1.9 

2.3 

2.0 

2.2 

2.0 

2.2 

2.0 

2.2 

2.1 

2.1 

2.1 

4 

3.1 

2.5 

3.1 

2.6 

3.0 

2.6 

3.0 

2.7 

2.9 

2.7 

2.9 

2.8 

2.8 

2.8 

5 

3.9 

3.1 

3.8 

3.2 

3.8 

3.3 

3.7 

3.3 

3.7 

3.4 

3.6 

3.5 

3.5 

3.5 

6 

4.7 

3.8 

4.6 

3.9 

4.5 

3.9 

4.5 

4.0 

4.4 

4.1 

4.3 

4.2 

4.2 

4.2 

7 

5.4 

4.4 

5.4 

4.5 

5.3 

4.6 

5.2 

4.7 

5.1 

4.8 

5.0 

4.9 

4.9 

4.9 

8 

6.2 

5.0 

6.1 

5.1 

6.0 

5.2 

5.9 

5.4 

5.9 

5.5 

5.8 

5.6 

5.7 

5.7 

9 

7.0 

5.7 

6.9 

5.8 

6.8 

5.9 

6.7 

6.0 

6.6 

6.1 

6.5 

6.3 

6.4 

6.4 

10 

7.8 

6.3 

7.7 

6.4 

7.5 

6.6 

7.4 

6.7 

7.3 

6.8 

7.2 

6.9 

7.1 

7.1 

11 

8.5 

6.9 

8.4 

7.1 

8.3 

7.2 

8.2 

7.4 

8.0 

7.5 

7.9 

7.6 

7.8 

7.8 

12 

9.3 

7.6 

9.2 

7.7 

9.1 

7.9 

8.9 

8.0 

8.8 

8.2 

8.6 

8.3 

8.5 

8.5 

13 

10.1 

8.2 

10.0 

8.4 

9.8 

8.5 

9.7 

8.7 

9.5 

8.9 

9.4 

9.0 

9.2 

9.2 

14 

10.9 

8.8 

10.7 

9.0 

10.6 

9.2 

10.4 

9.4 

10.2 

9.5 

10.1 

9.7 

9.9 

9.9 

15 

11.7 

9.4 

11.5 

9.6 

11.3 

9.8 

11.1 

10.0 

11.0 

10.2 

10.8 

10.4 

10.6 

10.6 

16 

12.4 

10.1 

12.3 

10.3 

12.1 

10.5 

11.9 

10.7 

11.7 

10.9 

11.5 

11.1 

11.3 

11.3 

17 

13.2 

10.7 

13.0 

10.9 

12.8 

11.2 

12.6 

11.4 

12.4 

11.6 

12.2 

11.8 

12.0 

12.0 

18 

14.0 

11.3 

13.8 

11.6 

13.6 

11.8 

13.4 

12.0 

13.2 

12.3 

12.9 

12.5 

12.7 

12.7 

19 

14.8 

12.0 

14.6 

12.2 

14.3 

12.5 

14.1 

12.7 

13.9 

13.0 

13.7 

13.2 

13.4 

13.4 

20 

15.5 

12.6 

15.3 

12.9 

15.1 

13.1 

14.9 

13.4 

14.6 

13.6 

14.4 

13.9 

14.1 

14.1 

21 

16.3 

13.2 

16.1 

13.5 

15.8 

13.8 

15.6 

14.1 

15.4 

14.3 

15.1 

14.6 

14.8 

14.8 

22 

17.1 

13.8 

16.9 

14.1 

16.6 

14.4 

16.3 

14.7 

16.1 

15.0 

15.8 

15.3 

15.6 

15.6 

23 

17.9 

14.5 

17.6 

14.8 

17.4 

15.1 

17.1 

15.4 

16.8 

15.7 

16.5 

16.0 

16.3 

16.3 

24 

18.7 

15.1 

18.4 

15.4 

18.1 

15.7 

17.8 

16.1 

17.6 

16.4 

17.3 

16.7 

17.0 

17.0 

25 

19.4 

15.7 

19.2 

16.1 

18.9 

16.4 

18.6 

16.7 

18.3 

17.0 

18.0 

17.4 

17.7 

17.7 

26 

20.2 

16.4 

19.9 

16.7 

19.6 

17.1 

19.3 

17.4 

19.0 

17.7 

18.7 

18.1 

18.4 

18.4 

27 

21.0 

17.0 

20.7 

17.4 

20.4 

17.7 

20.1 

18.1 

19.7 

18.4 

19.4 

18.8 

19.1 

19.1 

28 

21.8 

17.6 

21.4 

18.0 

21.1 

18.4 

20.8 

18.7 

20.5 

19.1 

20.1 

19.5 

19.8 

19.8 

29 

22.5 

18.3 

22.2 

18.6 

21.9 

19.0 

21.6 

19.4 

21.2 

19.8 

20.9 

20.1 

20.5 

20.5 

30 

23.3 

18.9 

23.0 

19.3 

22.6 

19.7 

22.3 

20.1 

21.9 

20.5 

21.6 

20.8 

21.2 

21.2 

31 

24.1 

19.5 

23.7 

19.9 

23.4 

20.3 

23.0 

20.7 

22.7 

21.1 

22.3 

21.5 

21.9 

21.9 

32 

24.9 

20.1 

24.5 

20.6 

24.2 

21.0 

23.8 

21.4 

23.4 

21.8 

23.0 

22.2 

22.6 

22.6 

33 

25.6 

20.8 

25.3 

21.2 

24^9 

21.6 

24.5 

22.1 

24.1 

22.5 

23.7 

22.9 

23.3 

23.3 

34 

26.4 

21.4 

26.0 

21.9 

25.7 

22.3 

25.3 

22.8 

24.9 

23.2 

24.5 

23.6 

24.0 

24.0 

35 

27.2 

22.0 

26.8 

22.5 

26.4 

23.0 

26.0 

23.4 

25.6 

23.9 

25.2 

24.3 

24.7 

24.7 

36 

28.0 

22.7 

27.6 

23.1 

27.2 

23.6 

26.8 

24.1 

26.3 

24.6 

25.9 

25.0 

25.5 

25.5 

37 

28.8 

23.3 

28.3 

23.8 

27.9 

24.3 

27.5 

24.8 

27.1 

25.2 

26.6 

25.7 

26.2 

26.2 

38 

29.5 

23.9 

29.1 

24.4 

28.7 

24.9 

28.2 

25.4 

27.8 

25.9 

27.3 

26.4 

26.9 

26.9 

39 

30.3 

24.5 

29.9 

25.1 

29.4 

25.6 

29.0 

26.1 

28.5 

26.6 

28.1 

27.1 

27.6 

27.6 

40 

31.1 

25.2 

30.6 

25.7 

30.2 

26.2 

29.7 

26.8 

29.3 

27.3 

28.8 

27.8 

28.3 

28.3 

41 

31.9 

25.8 

31.4 

26.4 

30.9 

26.9 

30.5 

27.4 

30.0 

28.0 

29.5 

28.5 

29.0 

29.0 

42 

32.6 

26.4 

32.2 

27.0 

31.7 

27.6 

31.2 

28.1 

30.7 

28.6 

30.2 

29.2 

29.7 

29.7 

43 

33.4 

27.1 

32.9 

27.6 

32.5 

28.2 

32.0 

28.8 

31.4 

29.3 

30.9 

29.9 

30.4 

30.4 

44 

34.2 

27.7 

33.7 

28.3 

33.2 

28.9 

32.7 

29.4 

32.2 

30.0 

31.7 

30.6 

31.1 

31.1 

45 

35.0 

28.3 

34.5 

28.9 

34.0 

29.5 

33.4 

30.1 

32.9 

30.7 

32.4 

31.3 

31.8 

31.8 

46 

35.7 

28.9 

35.2 

29.6 

34.7 

30.2 

34.2 

30.8 

33.6 

31.4 

33.1 

32.0 

32.5 

32.5 

47 

36.5 

29.6 

36.0 

30.2 

35.5 

30.8 

34.9 

31.4 

34.4 

32.1 

33.8 

32.6 

33.2 

33.2 

48 

37.3 

30.2 

36.8 

30.9 

36.2 

31.5 

35.7 

32.1 

35.1 

32.7 

34.5 

33.3 

33.9 

33.9 

49 

38.1 

30.8 

37.5 

31.5 

37.0 

32.1 

36.4 

32.8 

35.8 

33.4 

35.2 

34.0 

34.6 

34.6 

50 

38.9 

31.5 

38.3 

32.1 

37.7 

32.8 

37.2 

33.5 

36.6 

34.1 

36.0 

34.7 

35.4 

35.4 

100 

77.7 

62.9 

76.6 

64.3 

75.5 

65.6 

74.3 

66.9 

73.1 

68.2 

71.9 

69.5 

70.7 

70.7 

200 

155.4 

125.9 

153.2 

128.6 

150.9 

131.2 

148.6 

133.8 

146.3 

136.4 

143.9 

138.9 

141.4 

141.4 

300 

233.1 

188.8 

229.8 

192.8 

226.4 

196.8 

222.9 

200.7 

219.4 

204.6 

215.8 

208.4 

212.1 

212.1 

400 

310.9 

251.7 

306.4 

257.1 

301.9 

262.4 

297.3 

267.7 

292.6 

272.8 

287.7 

277.9 

282.8 

282.8 

500 

388.6 

314.7 

383.0 

321.4 

377.3 

328.0 

371.6 

334.6 

365.7 

341.0 

359.7 

347.3 

353.5 

353.5 

Dep. 

(129°, 

Lat. 
231°, 

Dep. 

Lat. 

Dep. 
(131°, 

Lat. 
229°, 

Dep. 
(132° 

Lat. 

228°, 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

(130° 

230°, 

(133°,  227°, 

(134°,  226°, 

(135°,  225°, 

30 

3°) 

31 

0°) 

3] 

1° 

31 

2°) 

313°) 

314°) 

315°) 

4|  P 

t.  51° 

5( 

3° 

4 

9° 

4i  P 

t.  48° 

47° 

46° 

4  Pt.  45° 

The  4-Pt.  or  45°  Courses  are  :  N.E..  N.W.,  S.E.,  S.W. 


Table  1.    Traverse  Table 


167 


3|Pt.39° 

40° 

41° 

31  Ft.  42° 

43° 

44° 

4Pt 

.  45° 

DiST. 

(141°,  219°, 

(140°,  220°, 

(139°,  221°, 

(138°,  222°, 

(137°,  223°, 

(136°,  224°, 

(135° 

,  225°. 

321°) 

320°) 

319°) 

318°) 

317°) 

316°) 

315°)         1 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

51 

39.6 

32.1 

39.1 

32.8 

38.5 

33.5 

37.9 

34.1 

37.3 

34.8 

36.7 

35.4 

36.1 

36.1 

52 

40.4 

32.7 

39.8 

33.4 

39.2 

34.1 

38.6 

34.8 

38.0 

35.5 

37.4 

36.1 

36.8 

36.8 

53 

41.2 

33.4 

40.6 

34.1 

40.0 

34.8 

39.4 

35.5 

38.8 

36.1 

38.1 

36.8 

37.5 

37.5 

54 

42.0 

34.0 

41.4 

34.7 

40.8 

35.4 

40.1 

36.1 

39.5 

36.8 

38.8 

37.5 

38.2 

38.2 

55 

42.7 

34.6 

42.1 

35.4 

41.5 

36.1 

40.9 

36.8 

40.2 

37.5 

39.6 

38.2 

38.9 

38.9 

56 

43.5 

35.2 

42.9 

36.0 

42.3 

36.7 

41.6 

37.5 

41.0 

38.2 

40.3 

38.9 

39.6 

39.6 

57 

44.3 

35.9 

43.7 

36.6 

43.0 

37.4 

42.4 

38.1 

41.7 

38.9 

41.0 

39.6 

40.3 

40.3 

58 

45.1 

36.5 

44.4 

37.3 

43.8 

38.1 

43.1 

38.8 

42.4 

39.6 

41.7 

40.3 

41.0 

41.0 

59 

45.9 

37.1 

45.2 

37.9 

44.5 

38.7 

43.8 

39.5 

43.1 

40.2 

42.4 

41.0 

41.7 

41.7 

60 

46.6 

37.8 

46.0 

38.6 

45.3 

39.4 

44.6 

40.1 

43.9 

40.9 

43.2 

41.7 

42.4 

42.4 

61 

47.4 

38.4 

46.7 

39.2 

46.0 

40.0 

45.3 

40.8 

44.6 

41.6 

43.9 

42.4 

43.1 

43.1 

62 

48.2 

39.0 

47.5 

39.9 

46.8 

40.7 

46.1 

41.5 

45.3 

42.3 

44.6 

43.1 

43.8 

43.8 

63 

49.0 

39.6 

48.3 

40.5 

47.5 

41.3 

46.8 

42.2 

46.1 

43.0 

45.3 

43.8 

44.5 

44.5 

64 

49.7 

40.3 

49.0 

41.1 

48.3 

42.0 

47.6 

42.8 

46.8 

43.6 

46.0 

44.5 

45.3 

45.3 

65 

50.5 

40.9 

49.8 

41.8 

49.1 

42.6 

48.3 

43.5 

47.5 

44.3 

46.8 

45.2 

46.0 

46.0 

66 

51.3 

41.5 

50.6 

42.4 

49.8 

43.3 

49.0 

44.2 

48.3 

45.0 

47.5 

45.8 

46.7 

46.7 

67 

52.1 

42.2 

51.3 

43.1 

50.6 

44.0 

49.8 

44.8 

49.0 

45.7 

48.2 

46.5 

47.4 

47.4 

68 

52.8 

42.8 

52.1 

43.7 

51.3 

44.6 

50.5 

45.5 

49.7 

46.4 

48.9 

47.2 

48.1 

48.1 

69 

53.6 

43.4 

52.9 

44.4 

52.1 

45.3 

51.3 

46.2 

50.5 

47.1 

49.6 

47.9 

48.8 

48.8 

70 

54.4 

44.1 

53.6 

45.0 

52.8 

45.9 

52.0 

46.8 

51.2 

47.7 

50.4 

48.6 

49.5 

49.5 

71 

55.2 

44.7 

54.4 

45.6 

53.6 

46.6 

52.8 

47.5 

51.9 

48.4 

51.1 

49.3 

50.2 

50.2 

72 

56.0 

45.3 

55.2 

46.3 

54.3 

47.2 

53.5 

48.2 

52.7 

49.1 

51.8 

50.0 

50.9 

50.9 

73 

56.7 

45.9 

55.9 

46.9 

55.1 

47.9 

54.2 

48.8 

53.4 

49.8 

52.5 

50.7 

51.6 

51.6 

74 

57.5 

46.6 

56.7 

47.6 

55.8 

48.5 

55.0 

49.5 

54.1 

50.5 

53.2 

51.4 

52.3 

52.3 

75 

58.3 

47.2 

57.5 

48.2 

56.6 

49.2 

55.7 

50.2 

54.9 

51.1 

54.0 

52.1 

53.0 

53.0 

76 

59.1 

47.8 

58.2 

48.9 

57.4 

49.9 

56.5 

50.9 

55.6 

51.8 

54.7 

52.8 

53.7 

53.7 

77 

59.8 

48.5 

59.0 

49.5 

58.1 

50.5 

57.2 

51.5 

56.3 

52.5 

55.4 

53.5 

54.4 

54.4 

78 

60.6 

49.1 

59.8 

50.1 

58.9 

51.2 

58.0 

52.2 

57.0 

53.2 

56.1 

54.2 

55.2 

55.2 

79 

61.4 

49.7 

60.5 

50.8 

59.6 

51.8 

58.7 

52.9 

57.8 

53.9 

56.8 

54.9 

55.9 

55.9 

80 

62.2 

50.3 

61.3 

51.4 

60.4 

52.5 

59.5 

53.5 

58.5 

54.6 

57.5 

55.6 

56.6 

56.6 

81 

62.9 

51.0 

62.0 

52.1 

61.1 

53.1 

60.2 

54.2 

59.2 

55.2 

58.3 

56.3 

57.3 

57.3 

82 

63.7 

51.6 

62.8 

52.7 

61.9 

53.8 

60.9 

54.9 

60.0 

55.9 

59.0 

57.0 

58.0 

58.0 

83 

64.5 

52.2 

63.6 

53.4 

62.6 

54.5 

61.7 

55.5 

60.7 

56.6 

59.7 

57.7 

58.7 

58.7 

84 

65.3 

52.9 

64.3 

54.0 

63.4 

55.1 

62.4 

56.2 

61.4 

57.3 

60.4 

58.4 

59.4 

59.4 

85 

66.1 

53.5 

65.1 

54.6 

64.2 

55.8 

63.2 

56.9 

62.2 

58.0 

61.1 

59.0 

60.1 

60.1 

86 

66.8 

54.1 

65.9 

55.3 

64.9 

56.4 

63.9 

57.5 

62.9 

58.7 

61.9 

59.7 

60.8 

60.8 

87 

67.6 

54.8 

66.6 

55.9 

65.7 

57.1 

64.7 

58.2 

63.6 

59.3 

62.6 

60.4 

61.5 

61.5 

88 

68.4 

55.4 

67.4 

56.6 

66.4 

57.7 

65.4 

58.9 

64.4 

60.0 

63.3 

61.1 

62.2 

62.2 

89 

69.2 

56.0 

68.2 

57.2 

67.2 

58.4 

66.1 

59.6 

65.1 

60.7 

64.0 

61.8 

62.9 

62.9 

90 

69.9 

56.6 

68.9 

57.9 

67.9 

59.0 

66.9 

60.2 

65.8 

61.4 

64.7 

62.5 

63.6 

63.6 

91 

70.7 

57.3 

69.7 

58.5 

68.7 

59.7 

67.6 

60.9 

66.6 

62.1 

65.5 

63.2 

64.3 

64.3 

92 

71.5 

57.9 

70.5 

59.1 

69.4 

60.4 

68.4 

61.6 

67.3 

62.7 

66.2 

63.9 

65.1 

65.1 

93 

72.3 

58.5 

71.2 

59.8 

70.2 

61.0 

69.1 

62.2 

68.0 

63.4 

66.9 

64.6 

65.8 

65.8 

94 

73.1 

59.2 

72.0 

60.4 

70.9 

61.7 

69.9 

62.9 

68.7 

64.1 

67.6 

65.3 

66.5 

66.5 

95 

73.8 

59.8 

72.8 

61.1 

71.7 

62.3 

70.6 

63.6 

69.5 

64.8 

68.3 

66.0 

67.2 

67.2 

96 

74.6 

60.4 

73.5 

61.7 

72.5 

63.0 

71.3 

64.2 

70.2 

65.5 

69.1 

66.7 

67.9 

67.9 

97 

75.4 

61.0 

74.3 

62.4 

73.2 

63.6 

72.1 

64.9 

70.9 

66.2 

69.8 

67.4 

68.6 

68.6 

98 

76.2 

61.7 

75.1 

63.0 

74.0 

64.3 

72.8 

65.6 

71.7 

66.8 

70.5 

68.1 

69.3 

69.3 

99 

76.9 

62.3 

75.8 

63.6 

74.7 

64.9 

73.6 

66.2 

72.4 

67.5 

71.2 

68.8 

70.0 

70.0 

100 

77.7 

62.9 

76.6 

64.3 

75.5 

65.6 

74.3 

66.9 

73.1 

68.2 

71.9 

69.5 

70.7 

70.7 

600 

466.3 

377.6 

459.6 

385.7 

452.8 

393.6 

445.9 

401.5 

438.8 

409.2 

431.6 

416.8 

424.3 

424.3 

700 

543.9 

440.6 

536.3 

450.0 

528.3 

459.2 

520.2 

468.4 

511.9 

477.4 

503.5 

486.3 

495.0 

495.0 

800 

621.8 

503.5 

613.0 

514.2 

603.9 

524.8 

594.6 

535.3 

585.1 

545.6 

575.4 

555.8 

565.7 

565.7 

900 

699.3 

566.3 

689.5 

578.5 

679.2 

590.3 

668.8 

602.2 

658.2 

613.8 

647.3 

625.2 

636.3 

636.3 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

Dep. 

Lat. 

(129°,  2.31°, 

(130°,  230°, 

(131°,  229°, 

(132°,  228°, 

(133°,  227°, 

(134°,  226°, 

(135° 

,  225°, 

309°) 

310°) 

311°) 

312°) 

313°) 

314°) 

31 

5°) 

41  Pt.  51° 

50° 

49° 

4i  Pt.  48° 

47° 

46° 

4  Pt 

.45° 

The  4-Pt.  or  45°  Courses  are :  N.E.,  N.W.,  S.E.,  S.W. 


168 


Table  2 


To  Change  Long.  Diff.  into  Dep.,  SUBTRACT  Tabular  Number 

FROM  Long.  Diff. 


Long. 
Diff. 

OR 

Dep. 

Middle  Latitude 

1° 

0.0 

2° 

0.0 

3° 

0.0 

4° 

5° 

6° 

7° 

8° 

9° 

10° 

11° 

12° 

0.0 

13° 

14° 

15° 

1 

0.0 

0.0 

0.0 

0.0 

0.0 

0.0 

0.0 

0.0 

0.0 

0.0 

0.0 

2 

0.0 

0.0 

0.0 

0.0 

0.0 

0.0 

0.0 

0.0 

0.0 

0.0 

0.0 

0.0 

0.1 

0.1 

0.1 

3 

0.0 

0.0 

0.0 

0.0 

0.0 

0.0 

0.0 

0.0 

0.0 

0.0 

0.1 

0.1 

0.1 

0.1 

0.1 

4 

0.0 

0.0 

0.0 

0.0 

0.0 

0.0 

0.0 

0.0 

0.0 

0.1 

0.1 

0.1 

0.1 

0.1 

0.1 

5 

0.0 

0.0 

0.0 

0.0 

0.0 

0.0 

0.0 

0.0 

0.1 

0.1 

0.1 

0.1 

0.1 

0.1 

0.2 

6 

0.0 

0.0 

0.0 

0.0 

0.0 

0.0 

0.0 

0.1 

0.1 

0.1 

0.1 

0.1 

0.2 

0.2 

0.2 

7 

0.0 

0.0 

0.0 

0.0 

0.0 

0.0 

0.1 

0.1 

0.1 

0.1 

0.1 

0.2 

0.2 

0.2 

0.2 

8 

0.0 

0.0 

0.0 

0.0 

0.0 

0.0 

0.1 

0.1 

0.1 

0.1 

0.1 

0.2 

0.2 

0.2 

0.3 

9 

0.0 

0.0 

0.0 

0.0 

0.0 

0.0 

0.1 

0.1 

0.1 

0.1 

0.2 

0.2 

0.2 

0.3 

0.3 

10 

0.0 

0.0 

0.0 

0.0 

0.0 

0.1 

0.1 

0.1 

0.1 

0.2 

0.2 

0.2 

0.3 

0.3 

0.3 

11 

0.0 

0.0 

0.0 

0.0 

0.0 

0.1 

0.1 

0.1 

0.1 

0.2 

0.2 

0.2 

0.3 

0.3 

0.4 

12 

0.0 

0.0 

0.0 

0.0 

0.0 

0.1 

0.1 

0.1 

0.1 

0.2 

0.2 

0.3 

0.3 

0.4 

0.4 

13 

0.0 

0.0 

0.0 

0.0 

0.0 

0.1 

0.1 

0.1 

0.2 

0.2 

0.2 

0.3 

0.3 

0.4 

0.4 

14 

0.0 

0.0 

0.0 

0.0 

0.1 

0.1 

0.1 

0.1 

0.2 

0.2 

0.3 

0.3 

0.4 

0.4 

0.5 

15 

0.0 

0.0 

0.0 

0.0 

0.1 

0.1 

0.1 

0.1 

0.2 

0.2 

0.3 

0.3 

0.4 

0.4 

0.5 

16 

0.0 

0.0 

0.0 

0.0 

0.1 

0.1 

0.1 

0.2 

0.2 

0.2 

0.3 

0.3 

0.4 

0.5 

0.5 

17 

0.0 

0.0 

0.0 

0.0 

0.1 

0.1 

0.1 

0.2 

0.2 

0.3 

0.3 

0.4 

0.4 

0.5 

0.6 

18 

0.0 

0.0 

0.0 

0.0 

0.1 

0.1 

0.1 

0.2 

0.2 

0.3 

0.3 

0.4 

0.5 

0.5 

0.6 

19 

0.0 

0.0 

0.0 

0.0 

0.1 

0.1 

0.1 

0.2 

0.2 

0.3 

0.3 

0.4 

0.5 

0.6 

0.6 

20 

0.0 

0.0 

0.0 

0.0 

0.1 

0.1 

0.1 

0.2 

0.2 

0.3 

0.4 

0.4 

0.5 

0.6 

0.7 

21 

0.0 

0.0 

0.0 

0.1 

0.1 

0.1 

0.2 

0.2 

0.3 

0.3 

0.4 

0.5 

0.5 

0.6 

0.7 

22 

0.0 

0.0 

0.0 

0.1 

0.1 

0.1 

0.2 

0.2 

0.3 

0.3 

0.4 

0.5 

0.6 

0.7 

0.7 

23 

0.0 

0.0 

0.0 

0.1 

0.1 

0.1 

0.2 

0.2 

0.3 

0.3 

0.4 

0.5 

0.6 

0.7 

0.8 

24 

0.0 

0.0 

0.0 

0.1 

0.1 

0.1 

0.2 

0.2 

0.3 

0.4 

0.4 

0.5 

0.6 

0.7 

0.8 

25 

0.0 

0.0 

0.0 

0.1 

0.1 

0.1 

0.2 

0.2 

0.3 

0.4 

0.5 

0.5 

0.6 

0.7 

0.9 

26 

0.0 

0.0 

0.0 

0.1 

0.1 

0.1 

0.2 

0.3 

0.3 

0.4 

0.5 

0.6 

0.7 

0.8 

0.9 

27 

0.0 

0.0 

0.0 

0.1 

0.1 

0.1 

0.2 

0.3 

0.3 

0.4 

0.5 

0.6 

0.7 

0.8 

0.9 

28 

0.0 

0.0 

0.0 

0.1 

0.1 

0.2 

0.2 

0.3 

0.3 

0.4 

0.5 

0.6 

0.7 

0.8 

1.0 

29 

0.0 

0.0 

0.0 

0.1 

0.1 

0.2 

0.2 

0.3 

0.4 

0.4 

0.5 

0.6 

0.7 

0.9 

1.0 

30 

0.0 

0.0 

0.0 

0.1 

0.1 

0.2 

0.2 

0.3 

0.4 

0.5 

0.6 

0.7 

0.8 

0.9 

1.0 

31 

0.0 

0.0 

0.0 

0.1 

0.1 

0.2 

0.2 

0.3 

0.4 

0.5 

0.6 

0.7 

0.8 

0.9 

1.1 

32 

0.0 

0.0 

0.0 

0.1 

0.1 

0.2 

0.2 

0.3 

0.4 

0.5 

0.6 

0.7 

0.8 

1.0 

1.1 

33 

0.0 

0.0 

0.0 

0.1 

0.1 

0.2 

0.2 

0.3 

0.4 

0.5 

0.6 

0.7 

0.8 

1.0 

1.1 

34 

0.0 

0.0 

0.0 

0.1 

0.1 

0.2 

0.3 

0.3 

0.4 

0.5 

0.6 

0.7 

0.9 

1.0 

1.2 

35 

0.0 

0.0 

0.0 

0.1 

0.1 

0.2 

0.3 

0.3 

0.4 

0.5 

0.6 

0.8 

0.9 

1.0 

1.2 

36 

0.0 

0.0 

0.0 

0.1 

0.1 

0.2 

0.3 

0.4 

0.4 

0.5 

0.7 

0.8 

0.9 

1.1 

1.2 

37 

0.0 

0.0 

0.1 

0.1 

0.1 

0.2 

0.3 

0.4 

0.5 

0.6 

0.7 

0.8 

0.9 

1.1 

1.3 

38 

0.0 

0.0 

0.1 

0.1 

0.1 

0.2 

0.3 

0.4 

0.5 

0.6 

0.7 

0.8 

1.0 

1.1 

1.3 

39 

0.0 

0.0 

0.1 

0.1 

0.1 

0.2 

0.3 

0.4 

0.5 

0.6 

0.7 

0.9 

1.0 

1.2 

1.3 

40 

0.0 

0.0 

0.1 

0.1 

0.2 

0.2 

0.3 

0.4 

0.5 

0.6 

0.7 

0.9 

1.0 

1.2 

1.4 

41 

0.0 

0.0 

0.1 

0.1 

0.2 

0.2 

0.3 

0.4 

0.5 

0.6 

0.8 

0.9 

1.1 

1.2 

1.4 

42 

0.0 

0.0 

0.1 

0.1 

0.2 

0.2 

0.3 

0.4 

0.5 

0.6 

0.8 

0.9 

1.1 

1.2 

1.4 

43 

0.0 

0.0 

0.1 

0.1 

0.2 

0.2 

0.3 

0.4 

0.5 

0.7 

0.8 

0.9 

1.1 

1.3 

1.5 

44 

0.0 

0.0 

0.1 

0.1 

0.2 

0.2 

0.3 

0.4 

0.5 

0.7 

0.8 

1.0 

1.1 

1.3 

1.5 

45 

0.0 

0.0 

0.1 

0.1 

0.2 

0.2 

0.3 

0.4 

0.6 

0.7 

0.8 

1.0 

1.2 

1.3 

1.5 

46 

0.0 

0.0 

0.1 

0.1 

0.2 

0.3 

0.3 

0.4 

0.6 

0.7 

0.8 

1.0 

1.2 

1.4 

1.6 

47 

0.0 

0.0 

0.1 

0.1 

0.2 

0.3 

0.4 

0.5 

0.6 

0.7 

0.9 

1.0 

1.2 

1.4 

1.6 

48 

0.0 

0.0 

0.1 

0.1 

0.2 

0.3 

0.4 

0.5 

0.6 

0.7 

0.9 

1.0 

1.2 

1.4 

1.6 

49 

0.0 

0.0 

0.1 

0.1 

0.2 

0.3 

0.4 

0.5 

0.6 

0.7 

0.9 

1.1 

1.3 

1.5 

1.7 

50 

0.0 

0.0 

0.1 

0.1 

0.2 

0.3 

0.4 

0.5 

0.6 

0.8 

0.9 

1.1 

1.3 

1.5 

1.7 

100 

0.0 

0.1 

0.1 

0.2 

0.4 

0.5 

0.7 

1.0 

1.2 

1.5 

1.8 

2.2 

2.6 

3.0 

3.4 

200 

0.0 

0.1 

0.3 

0.5 

0.8 

1.1 

1.5 

1.9 

2.5 

3.0 

3.7 

4.4 

5.1 

5.9 

6.8 

300 

0.0 

0.2 

0.4 

0.7 

1.1 

1.6 

2.2 

2.9 

3.7 

4.6 

5.5 

6.6 

7.7 

8.9 

10.2 

400 

0.1 

0.2 

0.6 

1.0 

1.5 

2.2 

3.0 

3.9 

4.9 

6.1 

7.4 

8.7 

10.2 

11.9 

13.7 

500 

0.1 

0.3 

0.7 

1.2 

1.9 

2.7 

3.7 

4.9 

6.2 

7.6 

9.2 

10.9 

12.8 

14.9 
1.03 

17.0 
1.04 

1.00 

1.00 

1.00 

1.00 

1.00 

1.01 

1.01 

1.01 

1.01 

1.02 

1.02 

1.02 

1.03 

I 

"'actoi 

a 

To  Change  Dep.  into  Long.  Diff.,  Multiply  Tabular  Number  by 
Factor  at  Foot  of  Column,  and  add  Product  to  Dep. 


Table  2 


169 


To  Change  Long.  Diff.  into  Dep.  subtract  Tabular  Number 

FROM  Long.  Diff. 


Long. 
Diff. 

OR 

Dep. 

Middle  Latitude 

1° 

2° 

3° 

J*l 

5° 

6° 

7° 

8° 

9° 

10° 

11° 

12° 

13° 

14° 

15° 

1.7 

51 

0.0 

0.0 

0.1 

0.1 

0.2 

0.3 

0.4 

0.5 

0.6 

0.8 

0.9 

1.1 

1.3 

1.5 

52 

0.0 

0.0 

0.1 

0.1 

0.2 

0.3 

0.4 

0.5 

0.6 

0.8 

1.0 

1.1 

1.3 

1.5 

1.8 

53 

0.0 

0.0 

0.1 

0.1 

0.2 

0.3 

0.4 

0.5 

0.7 

0.8 

1.0 

1.2 

1.4 

1.6 

1.8 

54 

0.0 

0.0 

0.1 

0.1 

0.2 

0.3 

0.4 

0.5 

0.7 

0.8 

1.0 

1.2 

1.4 

1.6 

1.8 

55 

0.0 

0.0 

0.1 

0.1 

0.2 

0.3 

0.4 

0.5 

0.7 

0.8 

1.0 

1.2 

1.4 

1.6 

1.9 

56 

0.0 

0.0 

0.1 

0.1 

0.2 

0.3 

0.4 

0.5 

0.7 

0.9 

1.0 

1.2 

1.4 

1.7 

1.9 

57 

0.0 

0.0 

0.1 

0.1 

0.2 

0.3 

0.4 

0.6 

0.7 

0.9 

1.0 

1.2 

1.5 

1.7 

1.9 

58 

0.0 

0.0 

0.1 

0.1 

0.2 

0.3 

0.4 

0.6 

0.7 

0.9 

1.1 

1.3 

1.5 

1.7 

2.0 

59 

0.0 

0.0 

0.1 

0.1 

0.2 

0.3 

0.4 

0.6 

0.7 

0.9 

1.1 

1.3 

1.5 

1.8 

2.0 

60 

0.0 

0.0 

0.1 

0.1 

0.2 

0.3 

0.4 

0.6 

0.7 

0.9 

1.1 

1.3 

1.5 

1.8 

2.0 

61 

0.0 

0.0 

0.1 

0.1 

0.2 

0.3 

0.5 

0.6 

0.8 

0.9 

1.1 

1.3 

1.6 

1.8 

2.1 

62 

0.0 

0.0 

0.1 

0.2 

0.2 

0.3 

0.5 

0.6 

0.8 

0.9 

1.1 

1.4 

1.6 

1.8 

2.1 

63 

0.0 

0.0 

0.1 

0.2 

0.2 

0.3 

0.5 

0.6 

0.8 

1.0 

1.2 

1.4 

1.6 

1.9 

2.1 

64 

0.0 

0.0 

0.1 

0.2 

0.2 

0.4 

0.5 

0.6 

0.8 

1.0 

1.2 

1.4 

1.6 

1.9 

2.2 

65 

0.0 

0.0 

0.1 

0.2 

0.2 

0.4 

0.5 

0.6 

0.8 

1.0 

1.2 

1.4 

1.7 

1.9 

2.2 

66 

0.0 

0.0 

0.1 

0.2 

0.3 

0.4 

0.5 

0.6 

0.8 

1.0 

1.2 

1.4 

1.7 

2.0 

2.2 

67 

0.0 

0.0 

0.1 

0.2 

0.3 

0.4 

0.5 

0.7 

0.8 

1.0 

1.2 

1.5 

1.7 

2.0 

2.3 

68 

0.0 

0.0 

0.1 

0.2 

0.3 

0.4 

0.5 

0.7 

0.8 

1.0 

1.2 

1.5 

1.7 

2.0 

2.3 

69 

0.0 

0.0 

0.1 

0.2 

0.3 

0.4 

0.5 

0.7 

0.8 

1.0 

1.3 

1.5 

1.8 

2.0 

2.4 

70 

0.0 

0.0 

0.1 

0.2 

0.3 

0.4 

0.5 

0.7 

0.9 

1.1 

1.3 

1.5 

1.8 

2.1 

2.4 

71 

0.0 

0.0 

0.1 

0.2 

0.3 

0.4 

0.5 

0.7 

0.9 

1.1 

1.3 

1.6 

1.8 

2.1 

2.4 

72 

0.0 

0.0 

0.1 

0.2 

0.3 

0.4 

0.5 

0.7 

0.9 

1.1 

1.3 

1.6 

1.8 

2.1 

2.5 

73 

0.0 

0.0 

0.1 

0.2 

0.3 

0.4 

0.5 

0.7 

0.9 

1.1 

1.3 

1.6 

1.9 

2.2 

2.5 

74 

0.0 

0.0 

0.1 

0.2 

0.3 

0.4 

0.6 

0.7 

0.9 

1.1 

1.4 

1.6 

1.9 

2.2 

2.5 

75 

0.0 

0.0 

0.1 

0.2 

0.3 

0.4 

0.6 

0.7 

0.9 

1.1 

1.4 

1.6 

1.9 

2.2 

2.6 

76 

0.0 

0.0 

0.1 

0.2 

0.3 

0.4 

0.6 

0.7 

0.9 

1.2 

1.4 

1.7 

1.9 

2.3 

2.6 

77 

0.0 

0.0 

0.1 

0.2 

0.3 

0.4 

0.6 

0.7 

0.9 

1.2 

1.4 

1.7 

2.0 

2.3 

2.6 

78 

0.0 

0.0 

0.1 

0.2 

0.3 

0.4 

0.6 

0.8 

1.0 

1.2 

1.4 

1.7 

2.0 

2.3 

2.7 

79 

0.0 

0.0 

0.1 

0.2 

0.3 

0.4 

0.6 

0.8 

1.0 

1.2 

1.5 

1.7 

2.0 

2.3 

2.7 

80 

0.0 

0.0 

0.1 

0.2 

0.3 

0.4 

0.6 

0.8 

1.0 

1.2 

1.5 

1.7 

2.1 

2.4 

2.7 

81 

0.0 

0.0 

0.1 

0.2 

0.3 

0.4 

0.6 

0.8 

1.0 

1.2 

1.5 

1.8 

2.1 

2.4 

2.8 

82 

0.0 

0.0 

0.1 

0.2 

0.3 

0.4 

0.6 

0.8 

1.0 

1.2 

1.5 

1.8 

2.1 

2.4 

2.8 

83 

0.0 

0.1 

0.1 

0.2 

0.3 

0.5 

0.6 

0.8 

1.0 

1.3 

1.5 

1.8 

2.1 

2.5 

2.8 

84 

0.0 

0.1 

0.1 

0.2 

0.3 

0.5 

0.6 

0.8 

1.0 

1.3 

1.5 

1.8 

2.2 

2.5 

2.9 

85 

0.0 

0.1 

0.1 

0.2 

0.3 

0.5 

0.6 

0.8 

1.0 

1.3 

1.6 

1.9 

2.2 

2.5 

2.9 

86 

0.0 

0.1 

0.1 

0.2 

0.3 

0.5 

0.6 

0.8 

1.1 

1.3 

1.6 

1.9 

2.2 

2.6 

2.9 

87 

0.0 

0.1 

0.1 

0.2 

0.3 

0.5 

0.6 

0.8 

1.1 

1.3 

1.6 

1.9 

2.2 

2.6 

3.0 

88 

0.0 

0.1 

0.1 

0.2 

0.3 

0.5 

0.7 

0.9 

1.1 

1.3 

1.6 

1.9 

2.3 

2.6 

3.0 

89 

0.0 

0.1 

0.1 

0.2 

0.3 

0.5 

0.7 

0.9 

1.1 

1.4 

1.6 

1.9 

2.3 

2.6 

3.0 

90 

0.0 

0.1 

0.1 

0.2 

0.3 

0.5 

0.7 

0.9 

1.1 

1.4 

1.7 

2.0 

2.3 

2.7 

3.1 

91 

0.0 

0.1 

0.1 

0.2 

0.3 

0.5 

0.7 

0.9 

1.1 

1.4 

1.7 

2.0 

2.3 

2.7 

3.1 

92 

0.0 

0.1 

0.1 

0.2 

0.4 

0.5 

0.7 

0.9 

1.1 

1.4 

1.7 

2.0 

2.4 

2.7 

3.1 

93 

0.0 

0.1 

0.1 

0.2 

0.4 

0.5 

0.7 

0.9 

1.1 

1.4 

1.7 

2.0 

2.4 

2.8 

3.2 

94 

0.0 

0.1 

0.1 

0.2 

0.4 

0.5 

0.7 

0.9 

1.2 

1.4 

1.7 

2.1 

2.4 

2.8 

3.2 

95 

0.0 

0.1 

0.1 

0.2 

0.4 

0.5 

0.7 

0.9 

1.2 

1.4 

1.7 

2.1 

2.4 

2.8 

3.2 

96 

0.0 

0.1 

0.1 

0.2 

0.4 

0.5 

0.7 

0.9 

1.2 

1.5 

1.8 

2.1 

2.5 

2.9 

3.3 

97 

0.0 

0.1 

0.1 

0.2 

0.4 

0.5 

0.7 

0.9 

1.2 

1.5 

1.8 

2.1 

2.5 

2.9 

3.3 

98 

0.0 

0.1 

0.1 

0.2 

0.4 

0.5 

0.7 

1.0 

1.2 

1.5 

1.8 

2.1 

2.5 

2.9 

3.3 

99 

0.0 

0.1 

0.1 

0.2 

0.4 

0.5 

0.7 

1.0 

1.2 

1.5 

1.8 

2.2 

2.5 

2.9 

3.4 

100 

0.0 

0.1 

0.1 

0.2 

0.4 

0.5 

0.7 

1.0 

1.2 

1.5 

1.8 

2.2 

2.6 

3.0 

3.4 

600 

0.1 

0.4 

0.8 

1.4 

2.3 

3.3 

4.5 

5.8 

7.4 

9.1 

10.0 

13.1 

15.4 

17.8 

20.5 

700 

0.2 

0.5 

1.0 

1.8 

2.8 

3.9 

5.1 

6.7 

8.7 

10.5 

12.9 

15.3 

17.9 

20.8 

23.9 

800 

0.2 

0.5 

1.1 

2.0 

3.1 

4.4 

5.9 

7.7 

9.8 

12.1 

14.8 

17.5 

20.6 

23.8 

27.3 

900 

0.3 

0.7 

1.4 

2.4 

3.6 

5.0 

6.7 

8.7 

11.2 

13.7 

16.7 

19.8 

23.2 

26.8 

30.8 

1.00 

1.00 

1.00 

1.00 

1.00 

1.01 

1.01 

1.01 
Fa 

1.01 

CTOR 

1.02 

1.02 

1.02 

1.03 

1.03 

1.04 

To  Change  Dep.  into  Long.  Diff.  Multiply  Tabular  Number  by 
Factor  at  Foot  of  Column  and  add  Product  to  Dep. 


170 


Table  2 


To  Change  Long. 


DiFF.   INTO    Dep.,   SUBTRACT  TABULAR  NuMBER 

FROM  Long.  Diff. 


Long. 
Diff. 

OR 

Dep. 

Middle  Latitude 

16° 

17° 

18° 

19° 

20° 

21° 

22° 

23° 

24° 

25° 

26° 

27° 

28° 

1 

0.0 

0.0 

0.0 

~o7r 

0.1 

0.1 

0.1 

0.1 

0.1 

0.1 

0.1 

0.1 

0.1 

2 

0.1 

0.1 

0.1 

0.1 

0.1 

0.1 

0.1 

0.2 

0.2 

0.2 

0.2 

0.2 

0.2 

3 

0.1 

0.1 

0.1 

0.2 

0.2 

0.2 

0.2 

0.2 

0.3 

0.3 

0.3 

0.3 

0.4 

4 

0.2 

0.2 

0.2 

0.2 

0.2 

0.3 

0.3 

0.3 

0.3 

0.4 

0.4 

0.4 

0.5 

5 

0.2 

0.2 

0.2 

0.3 

0.3 

0.3 

0.4 

0.4 

0.4 

0.5 

0.5 

0.5 

0.6 

6 

0.2 

0.3 

0.3 

0.3 

0.4 

0.4 

0.4 

0.5 

0.5 

0.6 

0.6 

0.7 

0.7 

7 

0.3 

0.3 

0.3 

0.4 

0.4 

0.5 

0.5 

0.6 

0.6 

0.7 

0.7 

0.8 

0.8 

8 

0.3 

0.3 

0.4 

0.4 

0.5 

0.5 

0.6 

0.6 

0.7 

0.7 

0.8 

0.9 

0.9 

9 

0.3 

0.4 

0.4 

0.5 

0.5 

0.6 

0.7 

0.7 

0.8 

0.8 

0.9 

1.0 

1.1 

10 

0.4 

0.4 

0.5 

0.5 

0.6 

0.7 

0.7 

0.8 

0.9 

0.9 

1.0 

1.1 

1.2 

11 

0.4 

0.5 

0.5 

0.6 

0.7 

0.7 

0.8 

0.9 

1.0 

1.0 

1.1 

1.2 

1.3 

12 

0.5 

0.5 

0.6 

0.7 

0.7 

0.8 

0.9 

1.0 

1.0 

1.1 

1.2 

1.3 

1.4 

13 

0.5 

0.6 

0.6 

0.7 

0.8 

0.9 

0.9 

1.0 

1.1 

1.2 

1.3 

1.4 

1.5 

14 

0.5 

0.6 

0.7 

0.8 

0.8 

0.9 

1.0 

1.1 

1.2 

1.3 

1.4 

1.5 

1.6 

15 

0.6 

0.7 

0.7 

0.8 

0.9 

1.0 

1.1 

1.2 

1.3 

1.4 

1.5 

1.6 

1.8 

16 

0.6 

0.7 

0.8 

0.9 

1.0 

1.1 

1.2 

1.3 

1.4 

1.5 

1.6 

1.7 

1.9 

17 

0.7 

0.7 

0.8 

0.9 

1.0 

1.1 

1.2 

1.4 

1.5 

1.6 

1.7 

1.9 

2.0 

18 

0.7 

0.8 

0.9 

1.0 

1.1 

1.2 

1.3 

1.4 

1.6 

1.7 

1.8 

2.0 

2.1 

19 

0.7 

0.8 

0.9 

1.0 

1.1 

1.3 

1.4 

1.5 

1.6 

1.8 

1.9 

2.1 

2.2 

20 

0.8 

0.9 

1.0 

1.1 

1.2 

1.3 

1.5 

1.6 

1.7 

1.9 

2.0 

2.2 

2.3 

21 

0.8 

0.9 

1.0 

1.1 

1.3 

1.4 

1.5 

1.7 

1.8 

2.0 

2.1 

2.3 

2.5 

22 

0.9 

1.0 

1.1 

1.2 

1.3 

1.5 

1.6 

1.7 

1.9 

2.1 

2.2 

2.4 

2.6 

23 

0.9 

1.0 

1.1 

1.3 

1.4 

1.5 

1.7 

1.8 

2.0 

2.2 

2.3 

2.5 

2.7 

24 

0.9 

1.0 

1.2 

1.3 

1.4 

1.6 

1.7 

1.9 

2.1 

2.2 

2.4 

2.6 

2.8 

25 

1.0 

1.1 

1.2 

1.4 

1.5 

1.7 

1.8 

2.0 

2.2 

2.3 

2.5 

2.7 

2.9 

26 

1.0 

1.1 

1.3 

1.4 

1.6 

1.7 

1.9 

2.1 

2.2 

2.4 

2.6 

2.8 

3.0 

27 

1.0 

1.2 

1.3 

1.5 

1.6 

1.8 

2.0 

2.1 

2.3 

2.5 

2.7 

2.9 

3.2 

28 

1.1 

1.2 

1.4 

1.5 

1.7 

1.9 

2.0 

2.2 

2.4 

2.6 

2.8 

3.1 

3.3 

29 

1.1 

1.3 

1.4 

1.6 

1.7 

1.9 

2.1 

2.3 

2.5 

2.7 

2.9 

3.2 

3.4 

30 

1.2 

1.3 

1.5 

1.6 

1.8 

2.0 

2.2 

2.4 

2.6 

2.8 

3.0 

3.3 

3.5 

31 

1.2 

1.4 

1.5 

1.7 

1.9 

2.1 

2.3 

2.5 

2.7 

2.9 

3.1 

3.4 

3.6 

32 

1.2 

1.4 

1.6 

1.7 

1.9 

2.1 

2.3 

2.5 

2.8 

3.0 

3.2 

3.5 

3.7 

33 

1.3 

1.4 

1.6 

1.8 

2.0 

2.2 

2.4 

2.6 

2.9 

3.1 

3.3 

3.6 

3.9 

34 

1.3 

1.5 

1.7 

1.9 

2.1 

2.3 

2.5 

2.7 

2.9 

3.2 

3.4 

3.7 

4.0 

35 

1.4 

1.5 

1.7 

1.9 

2.1 

2.3 

2.5 

2.8 

3.0 

3.3 

3.5 

3.8 

4.1 

36 

1.4 

1.6 

1.8 

2.0 

2.2 

2.4 

2.6 

2.9 

3.1 

3.4 

3.6 

3.9 

4.2 

37 

1.4 

1.6 

1.8 

2.0 

2.2 

2.5 

2.7 

2.9 

3.2 

3.5 

3.7 

4.0 

4.3 

38 

1.5 

1.7 

1.9 

2.1 

2.3 

2.5 

2.8 

3.0 

3.3 

3.6 

3.8 

4.1 

4.4 

39 

1.5 

1.7 

1.9 

2.1 

2.4 

2.6 

2.8 

3.1 

3.4 

3.7 

3.9 

4.3 

4.6 

40 

1.5 

1.7 

2.0 

2.2 

2.4 

2.7 

2.9 

3.2 

3.5 

3.7 

4.0 

4.4 

4.7 

41 

1.6 

1.8 

2.0 

2.2 

2.5 

2.7 

3.0 

3.3 

3.5 

3.8 

4.1 

4.5 

4.8 

42 

1.6 

1.8 

2.1 

2.3 

2.5 

2.8 

3.1 

3.3 

3.6 

3.9 

4.3 

4.6 

4.9 

43 

1.7 

1.9 

2.1 

2.3 

2.6 

2.9 

3.1 

3.4 

3.7 

4.0 

4.4 

4.7 

5.0 

44 

1.7 

1.9 

2.2 

2.4 

2.7 

2.9 

3.2 

3.5 

3.8 

4.1 

4.5 

4.8 

5.2 

45 

1.7 

2.0 

2.2 

2.5 

2.7 

3.0 

3.3 

3.6 

3.9 

4.2 

4.6 

4.9 

5.3 

46 

1.8 

2.0 

2.3 

2.5 

2.8 

3.1 

3.3 

3.7 

4.0 

4.3 

4.7 

5.0 

5.4 

47 

1.8 

2.1 

2.3 

2.6 

2.8 

3.1 

3.4 

3.7 

4.1 

4.4 

4.8 

5.1 

5.5 

48 

1.9 

2.1 

2.3 

2.6 

2.9 

3.2 

3.5 

3.8 

4.1 

4.5 

4.9 

5.2 

5.6 

49 

1.9 

2.1 

2.4 

2.7 

3.0 

3.3 

3.6 

3.9 

4.2 

4.6 

5.0 

5.3 

5.7 

50 

1.9 

2.2 

2.4 

2.7 

3.0 

3.3 

3.6 

4.0 

4.3 

4.7 

5.1 

5.4 

5.9 

100 

3.9 

4.4 

4.9 

5.4 

6.0 

6.6 

7.3 

7.9 

8.6 

9.4 

10.1 

10.9 

11.7 

200 

7.7 

8.7 

9.8 

10.9 

12.1 

13.3 

14.6 

15.9 

17.3 

18.7 

20.2 

21.8 

23.4 

300 

11.6 

13.1 

14.7 

16.3 

18.1 

19.9 

21.8 

23.8 

25.9 

28.1 

30.4 

32.7 

35.1 

400 

15.5 

17.5 

19.6 

21.8 

24.1 

26.6 

29.1 

31.8 

34.6 

37.5 

40.5 

43.6 

46.9 

500 

19.4 

21.9 

24.5 

27.2 

30.1 

33.2 

36.4 

39.8 

43.2 

46.9 

50.6 

54.5 

58.5 

1.04 

1.05 

1.05 

1.06 

1.06 

1.07 

] 

1.08 

^ACTOI 

1.09 

1.09 

1.10 

1.11 

1.12 

1.13 

To  Change  Dep.  into  Long.  Diff.,  Multiply  Tabular  Number  by 
Factor  at  Foot  of  Column  and  add  Product  to  Dep. 


Table  2 


111 


To  Change  Long.  Diff.  into  Dep.  subteact  Tabular  Number 

FROM  Long.  Diff. 


Long. 
Diff. 

OR 

Dep. 

MiDD 

LE  Latitude 

16° 

17° 

18° 

19° 

20° 

21° 

22° 

23° 

24° 

25° 

26° 

27° 

28° 

51 

2.0 

2.2 

2.5 

2.8 

3.1 

3.4 

3.7 

4.1 

4.4 

4.8 

5.2 

5.6 

6.0 

52 

2.0 

2.3 

2.5 

2.8 

3.1 

3.5 

3.8 

4.1 

4.5 

4.9 

5.3 

5.7 

6.1 

53 

2.1 

2.3 

2.6 

2.9 

3.2 

3.5 

3.9 

4.2 

4.6 

5.0 

5.4 

5.8 

6.2 

54 

2.1 

2.4 

2.6 

2.9 

3.3 

3.6 

3.9 

4.3 

4.7 

5.1 

5.5 

5.9 

6.3 

55 

2.1 

2.4 

2.7 

3.0 

3.3 

3.7 

4.0 

4.4 

4.8 

5.2 

5.6 

6.0 

6.4 

56 

2.2 

2.4 

2.7 

3.1 

3.4 

3.7 

4.1 

4.5 

4.8 

5.2 

5.7 

6.1 

6.6 

57 

2.2 

2.5 

2.8 

3.1 

3.4 

3.8 

4.2 

4.5 

4.9 

5.3 

5.8 

6.2 

6.7 

58 

2.2 

2.5 

2.8 

3.2 

3.5 

3.9 

4.2 

4.6 

5.0 

5.4 

5.9 

6.3 

6.8 

59 

2.3 

2.6 

2.9 

3.2 

3.6 

3.9 

4.3 

4.7 

5.1 

5.5 

6.0 

6.4 

6.9 

60 

2.3 

2.6 

2.9 

3.3 

3.6 

4.0 

4.4 

4.8 

5.2 

5.6 

6.1 

6.5 

7.0 

61 

2.4 

2.7 

3.0 

3.3 

3.7 

4.1 

4.4 

4.8 

5.3 

5.7 

6.2 

6.6 

7.1 

62 

2.4 

2.7 

3.0 

3.4 

3.7 

4.1 

4.5 

4.9 

5.4 

5.8 

6.3 

6.8 

7.3 

63 

2.4 

2.8 

3.1 

3.4 

3.8 

4.2 

4.6 

5.0 

5.4 

5.9 

6.4 

6.9 

7.4 

64 

2.5 

2.8 

3.1 

3.5 

3.9 

4.3 

4.7 

5.1 

5.5 

6.0 

6.5 

7.0 

7.5 

65 

2.5 

2.8 

3.2 

3.5 

3.9 

4.3 

4.7 

5.2 

5.6 

6.1 

6.6 

7.1 

7.6 

66 

2.6 

2.9 

3.2 

3.6 

4.0 

4.4 

4.8 

5.2 

5.7 

6.2 

6.7 

7.2 

7.7 

67 

2.6 

2.9 

3.3 

3.7 

4.0 

4.5 

4.9 

5.3 

5.8 

6.3 

6.8 

7.3 

7.8 

68 

2.6 

3.0 

3.3 

3.7 

4.1 

4.5 

5.0 

5.4 

5.9 

6.4 

6.9 

7.4 

8.0 

69 

2.7 

3.0 

3.4 

3.8 

4.2 

4.6 

5.0 

5.5 

6.0 

6.5 

7.0 

7.5 

8.1 

70 

2.7 

3.1 

3.4 

3.8 

4.2 

4.6 

5.1 

5.6 

6.1 

6.6 

7.1 

7.6 

8.2 

71 

2.8 

3.1 

3.5 

3.9 

4.3 

4.7 

5.2 

5.6 

6.1 

6.7 

7.2 

7.7 

8.3 

72 

2.8 

3.1 

3.5 

3.9 

4.3 

4.8 

5.2 

5.7 

6.2 

6.7 

7.3 

7.8 

8.4 

73 

2.8 

3.2 

3.6 

4.0 

4.4 

4.8 

5.3 

5.8 

6.3 

6.8 

7.4 

8.0 

8.5 

74 

2.9 

3.2 

3.6 

4.0 

4.5 

4.9 

5.4 

5.9 

6.4 

6.9 

7.5 

8.1 

8.7 

75 

2.9 

3.3 

3.7 

4.1 

4.5 

5.0 

5.5 

6.0 

6.5 

7.0 

7.6 

8.2 

8.8 

76 

2.9 

3.3 

3.7 

4.1 

4.6 

5.0 

5.5 

6.0 

6.6 

7.1 

7.7 

8.3 

8.9 

77 

3.0 

3.4 

3.8 

4.2 

4.6 

5.1 

5.6 

6.1 

6.7 

7.2 

7.8 

8.4 

9.0 

78 

3.0 

3.4 

3.8 

4.2 

4.7 

5.2 

5.7 

6.2 

6.7 

7.3 

7.9 

8.5 

9.1 

79 

3.1 

3.5 

3.9 

4.3 

4.8 

5.2 

5.8 

6.3 

6.8 

7.4 

8.0 

8.6 

9.2 

80 

3.1 

3.5 

3.9 

4.4 

4.8 

5.3 

5.8 

6.4 

6.9 

7.5 

8.1 

8.7 

9.4 

81 

3.1 

3.5 

4.0 

4.4 

4.9 

5.4 

5.9 

6.4 

7.0 

7.6 

8.2 

8.8 

9.5 

82 

3.2 

3.6 

4.0 

4.5 

4.9 

5.4 

6.0 

6.5 

7.1 

7.7 

8.3 

8.9 

9.6 

83 

3.2 

3.6 

4.1 

4.5 

5.0 

5.5 

6.0 

6.6 

7.2 

7.8 

8.4 

9.0 

9.7 

84 

3.3 

3.7 

4.1 

4.6 

5.1 

5.6 

6.1 

6.7 

7.3 

7.9 

8.5 

9.2 

9.8 

85 

3.3 

3.7 

4.2 

4.6 

5.1 

5.6 

6.2 

6.8 

7.3 

8.0 

8.6 

9.3 

9.9 

86 

3.3 

3.8 

4.2 

4.7 

5.2 

5.7 

6.3 

6.8 

7.4 

8.1 

8.7 

9.4 

10.1 

87 

3.4 

3.8 

4.3 

4.7 

5.2 

5.8 

6.3 

6.9 

7.5 

8.2 

8.8 

9.5 

10.2 

88 

3.4 

3.8 

4.3 

4.8 

5.3 

5.8 

6.4 

7.0 

7.6 

8.2 

8.9 

9.6 

10.3 

89 

3.4 

3.9 

4.4 

4.8 

5.4 

5.9 

6.5 

7.1 

7.7 

8.3 

9.0 

9.7 

10.4 

90 

3.5 

3.9 

4.4 

4.9 

5.4 

6.0 

6.6 

7.2 

7.8 

8.4 

9.1 

9.8 

10.5 

91 

3.5 

4.0 

4.5 

5.0 

5.5 

6.0 

6.6 

7.2 

7.9 

8.5 

9.2 

9.9 

10.7 

92 

3.6 

4.0 

4.5 

5.0 

5.5 

6.1 

6.7 

7.3 

8.0 

8.6 

9.3 

10.0 

10.8 

93 

3.6 

4.1 

4.6 

5.1 

5.6 

6.2 

6.8 

7.4 

8.0 

8.7 

9.4 

10.1 

10.9 

94 

3.6 

4.1 

4.6 

5.1 

5.7 

6.2 

6.8 

7.5 

8.1 

8.8 

9.5 

10.2 

11.0 

95 

3.7 

4.2 

4.6 

5.2 

5.7 

6.3 

6.9 

7.6 

8.2 

8.9 

9.6 

10.4 

11.1 

96 

3.7 

4.2 

4.7 

5.2 

5.8 

6.4 

7.0 

7.6 

8.3 

9.0 

9.7 

10.5 

11.2 

97 

3.8 

4.2 

4.7 

5.3 

5.8 

6.4 

7.1 

7.7 

8.4 

9.1 

9.8 

10.6 

11.4 

98 

3.8 

4.3 

4.8 

5.3 

5.9 

6.5 

7.1 

7.8 

8.5 

9.2 

9.9 

10.7 

11.5 

99 

3.8 

4.3 

4.8 

5.4 

6.0 

6.6 

7.2 

7.9 

8.6 

9.3 

10.0 

10.8 

11.6 

100 

3.9 

4.4 

4.9 

5.4 

6.0 

6.6 

7.3 

7.9 

8.6 

9.4 

10.1 

10.9 

11.7 

600 

23.2 

26.2 

29.4 

32.7 

36.2 

39.9 

43.7 

47.7 

51.9 

56.2 

60.7 

65.4 

70.2 

700 

27.2 

30.6 

34.2 

38.1 

42.1 

46.4 

50.9 

55.7 

60.5 

65.5 

70.8 

76.3 

82.0 

800 

31.0 

35.0 

39.2 

43.5 

48.2 

53.1 

58.2 

63.6 

69.2 

74.9 

80.9 

87.1 

93.7 

900 

35.0 

39.4 

44.1 

49.1 

54.3 

59.7 

65.5 

71.7 

77.9 

84.4 

91.1 

98.1 

105.5 

1.04 

1.05 

1.06 

1.06 

1.06 

1.07 

1.08 
Facto 

1.09 

R 

1.10 

1.10 

1.11 

1.12 

1.13 

To  Change   Dep.  into   Long.  Diff.  Multiply  T  abular  Number  by 
Factor  at  Foot  of  Column,  and  add  Product  to  Dep. 


172 


Table  2 


To  Change  Long. 


DiFF.  INTO  Dep.,  subtract  Tabular  Number 
FROM  Long.  Diff. 


Long. 
Diff. 

OR 

Dep. 

Middle  Latitude 

29° 

30° 

31° 

32° 

33° 

34° 

35° 

36° 

37° 

38° 

39° 

40° 

1 

0.1 

0.1 

0.1 

0.2 

0.2 

0.2 

0.2 

0.2 

0.2 

0.2 

0.2 

0.2 

2 

0.3 

0.3 

0.3 

0.3 

0.3 

0.3 

0.4 

0.4 

0.4 

0.4 

0.4 

0.5 

3 

0.4 

0.4 

0.4 

0.5 

0.5 

0.5 

0.5 

0.6 

0.6 

0.6 

0.7 

0.7 

4 

0.5 

0.5 

0.6 

0.6 

0.6 

0.7 

0.7 

0.8 

0.8 

0.8 

0.9 

0.9 

5 

0.6 

0.7 

0.7 

0.8 

0.8 

0.9 

0.9 

1.0 

1.0 

1.1 

1.1 

1.2 

6 

0.8 

0.8 

0.9 

0.9 

1.0 

1.0 

1.1 

1.1 

1.2 

1.3 

1.3 

1.4 

7 

0.9 

0.9 

1.0 

1.1 

1.1 

1.2 

1.3 

1.3 

1.4 

1.5 

1.6 

1.6 

8 

1.0 

1.1 

1.1 

1.2 

1.3 

1.4 

1.4 

1.5 

1.6 

1.7 

1.8 

1.9 

9 

1.1 

1.2 

1.3 

1.4 

1.5 

1.5 

1.6 

1.7 

1.8 

1.9 

2.0 

2.1 

10 

1.3 

1.3 

1.4 

1.5 

1.6 

1.7 

1.8 

1.9 

2.0 

2.1 

2.2 

2.3 

11 

1.4 

1.5 

1.6 

1.7 

1.8 

1.9 

2.0 

2.1 

2.2 

2.3 

2.5 

2.6 

12 

1.5 

1.6 

1.7 

1.8 

1.9 

2.1 

2.2 

2.3 

2.4 

2.5 

2.7 

2.8 

13 

1.6 

1.7 

1.9 

2.0 

2.1 

2.2 

2.4 

2.5 

2.6 

2.8 

2.9 

3.0 

14 

1.8 

1.9 

2.0 

2.1 

2.3 

2.4 

2.5 

2.7 

2.8 

3.0 

3.1 

3.3 

15 

1.9 

2.0 

2.1 

2.3 

2.4 

2.6 

2.7 

2.9 

3.0 

3.2 

3.3 

3.5 

16 

2.0 

2.1 

2.3 

2.4 

2.6 

2.7 

2.9 

3.1 

3.2 

3.4 

3.6 

3.7 

17 

2.1 

2.3 

2.4 

2.6 

2.7 

2.9 

3.1 

3.2 

3.4 

3.6 

3.8 

4.0 

18 

2.3 

2.4 

2.6 

2.7 

2.9 

3.1 

3.3 

3.4 

3.6 

3.8 

4.0 

4.2 

19 

2.4 

2.5 

2.7 

2.9 

3.1 

3.2 

3.4 

3.6 

3.8 

4.0 

4.2 

4.4 

20 

2.5 

2.7 

2.9 

3.0 

3.2 

3.4 

3.6 

3.8 

4.0 

4.2 

4.5 

4.7 

21 

2.6 

2.8 

3.0 

3.2 

3.4 

3.6 

3.8 

4.0 

4.2 

4.5 

4.7 

4.9 

22 

2.8 

2.9 

3.1 

3.3 

3.5 

3.8 

4.0 

4.2 

4.4 

4.7 

4.9 

5.1 

23 

2.9 

3.1 

3.3 

3.5 

3.7 

3.9 

4.2 

4.4 

4.6 

4.9 

5.1 

5.4 

24 

3.0 

3.2 

3.4 

3.6 

3.9 

4.1 

4.3 

4.6 

4.8 

5.1 

5.3 

5.6 

25 

3.1 

3.3 

3.6 

3.8 

4.0 

4.3 

4.5 

4.8 

5.0 

5.3 

5.6 

5.8 

26 

3.3 

3.5 

3.7 

4.0 

4.2 

4.4 

4.7 

5.0 

5.2 

5.5 

5.8 

6.1 

27 

3.4 

3.6 

3.9 

4.1 

4.4 

4.6 

4.9 

5.2 

5.4 

5.7 

6.0 

6.3 

28 

3.5 

3.8 

4.0 

4.3 

4.5 

4.8 

5.1 

5.3 

5.6 

5.9 

6.2 

6.6 

29 

3.6 

3.9 

4.1 

4.4 

4.7 

5.0 

5.2 

5.5 

5.8 

6.1 

6.5 

6.8 

30 

3.8 

4.0 

4.3 

4.6 

4.8 

5.1 

'5.4 

5.7 

6.0 

6.4 

6.7 

7.0 

31 

3.9 

4.2 

4.4 

4.7 

5.0 

5.3 

5.6 

5.9 

6.2 

6.6 

6.9 

7.3 

32 

4.0 

4.3 

4.6 

4.9 

5.2 

5.5 

5.8 

6.1 

6.4 

6.8 

7.1 

7.5 

33 

4.1 

4.4 

4.7 

5.0 

5.3 

5.6 

6.0 

6.3 

6.6 

7.0 

7.4 

7.7 

34 

4.3 

4.6 

4.9 

5.2 

5.5 

5.8 

6.1 

6.5 

6.8 

7.2 

7.6 

8.0 

35 

4.4 

4.7 

5.0 

5.3 

5.6 

6.0 

6.3 

6.7 

7.0 

7.4 

7.8 

8.2 

36 

4.5 

4.8 

5.1 

5.5 

5.8 

6.2 

6.5 

6.9 

7.2 

7.6 

8.0 

8.4 

37 

4.6 

5.0 

5.3 

5.6 

6.0 

6.3 

6.7 

7.1 

7.5 

7.8 

8.2 

8.7 

38 

4.8 

5.1 

5.4 

5.8 

6.1 

6.5 

6.9 

7.3 

7.7 

8.1 

8.5 

8.9 

39 

4.9 

5.2 

5.6 

5.9 

6.3 

6.7 

7.1 

7.4 

7.9 

8.3 

8.7 

9.1 

40 

5.0 

5.4 

5.7 

'6.1 

6.5 

6.8 

7.2 

7.6 

8.1 

8.5 

8.9 

9.4 

41 

5.1 

5.5 

5.9 

6.2 

6.6 

7.0 

7.4 

7.8 

8.3 

8.7 

9.1 

9.6 

42 

5.3 

5.6 

6.0 

6.4 

6.8 

7.2 

7.6 

8.0 

8.5 

8.9 

9.4 

9.8 

43 

5.4 

5.8 

6.1 

6.5 

6.9 

7.4 

7.8 

8.2 

8.7 

9.1 

9.6 

10.1 

44 

5.5 

5.9 

6.3 

6.7 

7.1 

7.5 

8.0 

8.4 

8.9 

9.3 

9.8 

10.3 

45 

5.6 

6.0 

6.4 

6.8 

7.3 

7.7 

8.1 

8.6 

9.1 

9.5 

10.0 

10.5 

46 

5.8 

6.2 

6.6 

7.0 

7.4 

7.9 

8.3 

8.8 

9.3 

9.8 

10.3 

10.8 

47 

5.9 

6.3 

6.7 

7.1 

7.6 

8.0 

8.5 

9.0 

9.5 

10.0 

10.5 

11.0 

48 

6.0 

6.4 

6.9 

7.3 

7.7 

8.2 

8.7 

9.2 

9.7 

10.2 

10.7 

11.2 

49 

6.1 

6.6 

7.0 

7.4 

7.9 

8.4 

8.9 

9.4 

9.9 

10.4 

10.9 

11.5 

50 

6.3 

6.7 

7.1 

7.6 

8.1 

8.5 

9.0 

9.5 

10.1 

10.6 

11.1 

11.7 

100 

12.5 

13.4 

14.3 

15.2 

16.1 

17.1 

18.1 

19.1 

20.1 

21.2 

22.3 

23.4 

200 

25.1 

26.8 

28.6 

30.4 

32.3 

34.2 

36.2 

38.2 

40.3 

42.4 

44.6 

46.8 

300 

37.6 

40.2 

42.9 

45.6 

48.4 

51.3 

54.3 

57.3 

60.4 

63.6 

66.9 

70.2 

400 

50.2 

53.6 

57.1 

60.8 

64.5 

68.4 

72.3 

76.4 

80.6 

84.8 

89.1 

93.6 

500 

62.7 

67.0 

71.4 

76.0 

80.7 

85.5 

90.4 

95.5 

100.7 

106.0 

111.4 

117.0 

1.14 

1.15 

1.17 

1.18 

1.19 

1.21 

1.22 

1.24 

1.25 

1.27 

1.29 

1.31 

Factor 

To  Change  Dep.  into  Long.  Diff.,  Multiply  Tabular  Number   by 
Factor  at  Foot  of  Column,  and  add  Product  to  Dep. 


Table  2 


173 


To  Change  Long.  Diff.  into  Dep.  subteact  Tabular  Number 

FROM  Long.  Diff. 


Long. 
Diff. 

OR 

Dbp. 

Middle  Latitude 

29° 

30° 

31° 

32° 

33° 

34° 

35° 

36° 

37° 

38° 

39° 

40° 

51 

6.4 

6.8 

7.3 

7.7 

8.2 

8.7 

9.2 

9.7 

10.3 

10.8 

11.4 

11.9 

52 

6.5 

7.0 

7.4 

7.9 

8.4 

8.9 

9.4 

9.9 

10.5 

11.0 

11.6 

12.2 

53 

6.6 

7.1 

7.6 

8.1 

8.6 

9.1 

9.6 

10.1 

10.7 

11.2 

11.8 

12.4 

54 

6.8 

7.2 

7.7 

8.2 

8.7 

9.2 

9.8 

10.3 

10.9 

11.4 

12.0 

12.6 

55 

6.9 

7.4 

7.9 

8.4 

8.9 

9.4 

9.9 

10.5 

11.1 

11.7 

12.3 

12.9 

56 

7.0 

7.5 

8.0 

8.5 

9.0 

9.6 

10.1 

10.7 

11.3 

11.9 

12.5 

13.1 

57 

7.1 

7.6 

8.1 

8.7 

9.2 

9.7 

10.3 

10.9 

11.5 

12.1 

12.7 

13.3 

58 

7.3 

7.8 

8.3 

8.8 

9.4 

9.9 

10.5 

11.1 

11.7 

12.3 

12.9 

13.6 

59 

7.4 

7.9 

8.4 

9.0 

9.5 

10.1 

10.7 

11.3 

11.9 

12.5 

13.1 

13.8 

60 

7.5 

8.0 

8.6 

9.1 

9.7 

10.3 

10.9 

11.5 

12.1 

12.7 

13.4 

14.0 

61 

7.6 

8.2 

8.7 

9.3 

9.8 

10.4 

11.0 

11.6 

12.3 

12.9 

13.6 

14.3 

62 

7.8 

8.3 

8.9 

9.4 

10.0 

10.6 

11.2 

11.8 

12.5 

13.1 

13.8 

14.5 

63 

7.9 

8.4 

9.0 

9.6 

10.2 

10.8 

11.4 

12.0 

12.7 

13.4 

14.0 

14.7 

64 

8.0 

8.6 

9.1 

9.7 

10.3 

10.9 

11.6 

12.2 

12.9 

13.6 

14.3 

15.0 

65 

8.1 

8.7 

9.3 

9.9 

10.5 

11.1 

11.8 

12.4 

13.1 

13.8 

14.5 

15.2 

66 

8.3 

8.8 

9.4 

10.0 

10.6 

11.3 

11.9 

12.6 

13.3 

14.0 

14.7 

15.4 

67 

8.4 

9.0 

9.6 

10.2 

10.8 

11.5 

12.1 

12.8 

13.5 

14.2 

14.9 

15.7 

68 

8.5 

9.1 

9.7 

10.3 

11.0 

11.6 

12.3 

13.0 

13.7 

14.4 

15.2 

15.9 

69 

8.7 

9.2 

9.9 

10.5 

11.1 

11.8 

12.5 

13.2 

13.9 

14.6 

15.4 

16.1 

70 

8.8 

9.4 

10.0 

10.6 

11.3 

12.0 

12.7 

13.4 

14.1 

14.8 

15.6 

16.4 

71 

8.9 

9.5 

10.1 

10.8 

11.5 

12.1 

12.8 

13.6 

14.3 

15.1 

15.8 

16.6 

72 

9.0 

9.6 

10.3 

10.9 

11.6 

12.3 

13.0 

13.8 

14.5 

15.3 

16.0 

16.8 

73 

9.2 

9.8 

10.4 

11.1 

11.8 

12.5 

13.2 

13.9 

14.7 

15.5 

16.3 

17.1 

74 

9.3 

9.9 

10.6 

11.2 

11.9 

12.7 

13.4 

14.1 

14.9 

15.7 

16.5 

17.3 

75 

9.4 

10.0 

10.7 

11.4 

12.1 

12.8 

13.6 

14.3 

15.1 

15.9 

16.7 

17.5 

76 

9.5 

10.2 

10.9 

11.5 

12.3 

13.0 

13.7 

14.5 

15.3 

16.1 

16.9 

17.8 

77 

9.7 

10.3 

11.0 

11.7 

12.4 

13.2 

13.9 

14.7 

15.5 

16.3 

17.2 

18.0 

78 

9.8 

10.5 

11.1 

11.9 

12.6 

13.3 

14.1 

14.9 

15.7 

16.5 

17.4 

18.2 

79 

9.9 

10.6 

11.3 

12.0 

12.7 

13.5 

14.3 

15.1 

15.9 

16.7 

17.6 

18.5 

80 

10.0 

10.7 

11.4 

12.2 

12.9 

13.7 

14.5 

15.3 

16.1 

17.0 

17.8 

18.7 

81 

10.2 

10.9 

11.6 

12.3 

13.1 

13.8 

14.6 

15.5 

16.3 

17.2 

18.1 

19.0 

82 

10.3 

11.0 

11.7 

12.5 

13.2 

14.0 

14.8 

15.7 

16.5 

17.4 

18.3 

19.2 

83 

10.4 

11.1 

11.9 

12.6 

13.4 

14.2 

15.0 

15.9 

16.7 

17.6 

18.5 

19.4 

84 

10.5 

11.3 

12.0 

12.8 

13.6 

14.4 

15.2 

16.0 

16.9 

17.8 

18.7 

19.7 

85 

10.7 

11.4 

12.1 

12.9 

13.7 

14.5 

15.4 

16.2 

17.1 

18.0 

18.9 

19.9 

86 

10.8 

11.5 

12.3 

13.1 

13.9 

14.7 

15.6 

16.4 

17.3 

18.2 

19.2 

20.1 

87 

10.9 

11.7 

12.4 

13.2 

14.0 

14.9 

15.7 

16.6 

17.5 

18.4 

19.4 

20.4 

88 

11.0 

11.8 

12.6 

13.4 

14.2 

15.0 

15.9 

16.8 

17.7 

18.7 

19.6 

20.6 

89 

11.2 

11.9 

12.7 

13.5 

14.4 

15.2 

16.1 

17.0 

17.9 

18.9 

19.8 

20.8 

90 

11.3 

12.1 

12.9 

13.7 

14.5 

15.4 

16.3 

17.2 

18.1 

19.1 

20.1 

21.1 

91 

11.4 

12.2 

13.0 

13.8 

14.7 

15.6 

16.5 

17.4 

18.3 

19.3 

20.3 

21.3 

92 

11.5 

12.3 

13.1 

14.0 

14.8 

15.7 

16.6 

17.6 

18.5 

19.5 

20.5 

21.5 

93 

11.7 

12.5 

13.3 

14.1 

15.0 

15.9 

16.8 

17.8 

18.7 

19.7 

20.7 

21.8 

94 

11.8 

12.6 

13.4 

14.3 

15.2 

16.1 

17.0 

18.0 

18.9 

19.9 

20.9 

22.0 

95 

11.9 

12.7 

13.6 

14.4 

15.3 

16.2 

17.2 

18.1 

19.1 

20.1 

21.2 

22.2 

96 

12.0 

12.9 

13.7 

14.6 

15.5 

16.4 

17.4 

18.3 

19.3 

20.4 

21.4 

22.5 

97 

12.2 

13.0 

13.9 

14.7 

15.6 

16.6 

17.5 

18.5 

19.5 

20.6 

21.6 

22.7 

98 

12.3 

13.1 

14.0 

14.9 

15.8 

16.8 

17.7 

18.7 

19.7 

20.8 

21.8 

22.9 

99 

12.4 

13.3 

14.1 

15.0 

16.0 

16.9 

17.9 

18.9 

19.9 

21.0 

22.1 

23.2 

100 

12.5 

13.4 

14.3 

15.2 

16.1 

17.1 

18.1 

19.1 

20.1 

21.2 

22.3 

23.4 

600 

75.2 

80.4 

85.7 

91.2 

96.8 

102.6 

108.5 

114.6 

120.8 

127.2 

133.7 

140.4 

700 

87.8 

93.9 

99.9 

106.4 

113.0 

119.7 

126.5 

133.8 

141.0 

148.4 

156.1 

163.7 

800 

100.3 

107.2 

114.2 

121.6 

129.0 

136.7 

144.6 

152.7 

161.1 

169.6 

178.2 

187.0 

900 

113.0 

120.7 

128.6 

136.8 

145.2 

153.9 

162.8 

171.9 

181.4 

190.9 

200.7 

210.5 

1.14 

1.15 

1.17 

1.18 

1.19 

1.21 
Fac 

1.22 

TOR 

1.24 

1.25 

1.27 

1.29 

1.31 

To  Change  Dep.  into  Long.  Diff.  Multiply  Tabular  Number  by 
Factor  at  Foot  of  Column  and  ADD  Product  to  Dep. 


174 


Table  2 


To    Change    Long.   Diff.   into   Dep.,   subtract   Tabular   Number 

FROM  Long.  Diff. 


Long. 
Diff. 

OR 

Dep. 

Middle  Latitude 

41° 

42° 

43° 

44° 

45° 

46° 

47° 

48° 

49° 

50° 

51° 

1 

0.2 

0.3 

0.3 

0.3 

0.3 

0.3 

0.3 

0.3 

0.3 

0.4 

0.4 

2 

0.5 

0.5 

0.5 

0.6 

0.6 

0.6 

0.6 

0.7 

0.7 

0.7 

0.7 

3 

0.7 

0.8 

0.8 

0.8 

0.9 

0.9 

1.0 

1.0 

1.0 

1.1 

1.1 

4 

1.0 

1.0 

1.1 

1.1 

1.2 

1.2 

1.3 

1.3 

1.4 

1.4 

15 

5 

1.2 

1.3 

1.3 

1.4 

1.5 

1.5 

1.6 

1.7 

1.7 

1.8 

1.9 

6 

1.5 

1.5 

1.6 

1.7 

1.8 

1.8 

1.9 

2.0 

2.1 

2.1 

2.2 

7 

1.7 

1.8 

1.9 

2.0 

2.1 

2.1 

2.2 

2.3 

2.4 

2.5 

2.6 

8 

2.0 

2.1 

2.1 

2.2 

2.3 

2.4 

2.5 

2.6 

2.8 

2.9 

3.0 

9 

2.2 

2.3 

2.4 

2.5 

2.6 

2.7 

2.9 

3.0 

3.1 

3.2 

3.3 

10 

2.5 

2.6 

2.7 

2.8 

2.9 

3.1 

3.2 

3.3 

3.4 

3.6 

3.7 

11 

2.7 

2.8 

3.0 

3.1 

3.2 

3.4 

3.5 

3.6 

3.8 

3.9 

4.1 

12 

2.9 

3.1 

3.2 

3.4 

3.5 

3.7 

3.8 

4.0 

4.1 

4.3 

4.4 

13 

3.2 

3.3 

3.5 

3.6 

3.8 

4.0 

4.1 

4.3 

4.5 

4.6 

4.8 

14 

3.4 

3.6 

3.8 

3.9 

4.1 

4.3 

4.5 

4.6 

4.8 

5.0 

5.2 

15 

3.7 

3.9 

4.0 

4.2 

4.4 

4.6 

4.8 

5.0 

5.2 

5.4 

5.6 

16 

3.9 

4.1 

4.3 

4.5 

4.7 

4.9 

5.1 

5.3 

5.5 

5.7 

5.9 

17 

4.2 

4.4 

4.6 

4.8 

5.0 

5.2 

5.4 

5.6 

5.8 

6.1 

6.3 

18 

4.4 

4.6 

4.8 

5.1 

5.3 

5.5 

5.7 

6.0 

6.2 

6.4 

6.7 

19 

4.7 

4.9 

5.1 

5.3 

5.6 

5.8 

6.0 

6.3 

6.5 

6.8 

7.0 

20 

4.9 

5.1 

5.4 

5.6 

5.9 

6.1 

6.4 

6.6 

6.9 

7.1 

7.4 

21 

5.2 

5.4 

5.6 

5.9 

6.2 

6.4 

6.7 

6.9 

7.2 

7.5 

7.8 

22 

5.4 

5.7 

5.9 

6.2 

6.4 

6.7 

7.0 

7.3 

7.6 

7.9 

8.2 

23 

5.6 

5.9 

6.2 

6.5 

6.7 

7.0 

7.3 

7.6 

7.9 

8.2 

8.5 

24 

5.9 

6.2 

6.4 

6.7 

7.0 

7.3 

7.6 

7.9 

8.3 

8.6 

8.9 

25 

6.1 

6.4 

6.7 

7.0 

7.3 

7.6 

8.0 

8.3 

8.6 

8.9 

9.3 

26 

6.4 

6.7 

7.0 

7.3 

7.6 

7.9 

8.3 

8.6 

8.9 

9.3 

9.6 

27 

6.6 

6.9 

7.3 

7.6 

7.9 

8.2 

8.6 

8.9 

9.3 

9.6 

10.0 

28 

6.9 

7.2  , 

7.5 

7.9 

8.2 

8.5 

8.9 

9.3 

9.6 

10.0 

10.4 

29 

7.1 

7.4 

7.8 

8.1 

8.5 

8.9 

9.2 

9.6 

10.0 

10.4 

10.7 

30 

7.4 

7.7 

8.1 

8.4 

8.8 

9.2 

9.5 

9.9 

10.3 

10.7 

11.1 

31 

7.6 

8.0 

8.3 

8.7 

9.1 

'9.5 

9.9 

10.3 

10.7 

11.1 

11.5 

32 

7.8 

8.2 

8.6 

9.0 

9.4 

9.8 

10.2 

10.6 

11.0 

11.4 

11.9 

33 

8.1 

8.5 

8.9 

9.3 

9.7 

10.1 

10.5 

10.9 

11.4 

11.8 

12.2 

34 

8.3 

8.7 

9.1 

9.5 

10.0 

10.4 

10.8 

11.2 

11.7 

12.1 

12^6 

35 

8.6 

9.0 

9.4 

9.8 

10.3 

10.7 

11.1 

11.6 

12.0 

12.5 

13.0 

36 

8.8 

9.2 

9.7 

10.1 

10.5 

11.0 

11.4 

11.9 

12.4 

12.9 

13.3 

37 

9.1 

9.5 

9.9 

10.4 

10.8 

11.3 

11.8 

12.2 

12.7 

13.2 

13.7 

38 

9.3 

9.8 

10.2 

10.7 

11.1 

11.6 

12.1 

12.6 

13.1 

13.6 

14.1 

39 

9.6 

10.0 

10.5 

10.9 

11.4 

11.9 

12.4 

12.9 

13.4 

13.9 

14.5 

40 

9.8 

10.3 

10.7 

11.2 

11.7 

12.2 

12.7 

13.2 

13.8 

14.3 

14.8 

41 

10.1 

10.5 

11.0 

11.5 

12.0 

12.5 

13.0 

13.6 

14.1 

14.6 

15.2 

42 

10.3 

10.8 

11.3 

11.8 

12.3 

12.8 

13.4 

13.9 

14.4 

15.0 

15.6 

43 

10.5 

11.0 

11.6 

12.1 

12.6 

13.1 

13.7 

14.2 

14.8 

15.4 

15.9 

44 

10.8 

11.3 

11.8 

12.3 

12.9 

13.4 

14.0 

14.6 

15.1 

15.7 

16.3 

45 

11.0 

11.6 

12.1 

12.6 

13.2 

13.7 

14.3 

14.9 

15.5 

16.1 

16.7 

46 

11.3 

11.8 

12.4 

12.9 

13.5 

14.0 

14.6 

15.2 

15.8 

16.4 

17.1 

47 

11.5 

12.1 

12.6 

13.2 

13.8 

14.4 

14.9 

15.6 

16.2 

16.8 

17.4 

48 

11.8 

12.3 

12.9 

13.5 

14.1 

14.7 

15.3 

15.9 

16.5 

17.1 

17.8 

49 

12.0 

12.6 

13.2 

13.8 

14.4 

15.0 

15.6 

16.2 

16.9 

17.5 

18.2 

50 

12.3 

12.8 

13.4 

14.0 

14.6 

15.3 

15.9 

16.5 

17.2 

17.9 

18.5 

100 

24.5 

25.7 

26.9 

28.1 

29.3 

30.5 

31.8 

33.1 

34.4 

35.7 

37.1 

200 

49.1 

51.4 

53.7 

56.1 

58.6 

61.1 

63.6 

66.2 

68.8 

71.4 

74.1 

300 

73.6 

77.1 

80.6 

84.2 

87.9 

91.6 

95.4 

99.3 

103.2 

107.2 

111.2 

400 

98.1 

102.7 

107.4 

112.3 

117.2 

122.1 

127.2 

132.3 

137.6 

142.9 

148.3 

500 

122.7 

128.4 

134.3 

140.3 

146.5 

152.7 

159.0 

165.4 

172.0 

178.6 

185.3 

1.33 

1.35 

1.37 

1.39 

1.41 

1.44 
Factor 

1.47 

1.50 

1.52 

1.56 

1.59 

To  Change  Dep.  into  Long.  Diff.,  Multiply  Tabular  Number  by 
Factor  at  Foot  of  Column,  and  add  Product  to  Dep. 


Table  2 


175 


To  Change  Long. 


DiFF.  INTO  Dep.  subtract  Tabular  Number 
FROM  Long.  Diff. 


Long. 
Diff 

OR 

Dep. 

Middle  Latitude 

41° 

42° 

43° 

44° 

45° 

46° 

47° 

48° 

49° 

50° 

51° 

51 

12.5 

13.1 

13.7 

14.3 

14.9 

15.6 

16.2 

16.9 

17.5 

18.2 

18.9 

52 

12.8 

13.4 

14.0 

14.6 

15.2 

15.9 

16.5 

17.2 

17.9 

18.6 

19.3 

53 

13.0 

13.6 

14.2 

14.9 

15.5 

16.2 

16.9 

17.5 

18.2 

18.9 

19.6 

54 

13.2 

13.9 

14.5 

15.2 

15.8 

16.5 

17.2 

17.9 

18.6 

19.3 

20.0 

55 

13.5 

14.1 

14.8 

15.4 

16.1 

16.8 

17.5 

18.2 

18.9 

19.6 

20.4 

56 

13.7 

14.4 

15.0 

15.7 

16.4 

17.1 

17.8 

18.5 

19.3 

20.0 

20.8 

57 

14.0 

14.6 

15.3 

16.0 

16.7 

17.4 

18.1 

18.9 

19.6 

20.4 

21.1 

58 

14.2 

14.9 

15.6 

16.3 

17.0 

17.7 

18.4 

19.2 

19.9 

20.7 

21.5 

59 

14.5 

15.2 

15.9 

16.6 

17.3 

18.0 

18.8 

19.5 

20.3 

21.1 

21.9 

60 

14.7 

15.4 

16.1 

16.8 

17.6 

18.3 

19.1 

19.9 

20.6 

21.4 

22.2 

61 

15.0 

15.7 

16.4 

17.1 

17.9 

18.6 

19.4 

20.2 

21.0 

21.8 

22.6 

62 

15.2 

15.9 

16.7 

17.4 

18.2 

18.9 

19.7 

20.5 

21.3 

22.1 

23.0 

63 

15.5 

16.2 

16.9 

17.7 

18.5 

19.2 

20.0 

20.8 

21.7 

22.5 

23.4 

64 

15.7 

16.4 

17.2 

18.0 

18.7 

19.5 

20.4 

21.2 

22.0 

22.9 

23.7 

65 

15.9 

16.7 

17.5 

18.2 

19.0 

19.8 

20.7 

21.5 

22.4 

23.2 

24.1 

66 

16.2 

17.0 

17.7 

18.5 

19.3 

20.2 

21.0 

21.8 

22.7 

23.6 

24.5 

67 

16.4 

17.2 

18.0 

18.8 

19.6 

20.5 

21.3 

22.2 

23.0 

23.9 

24.8 

68 

16.7 

17.5 

18.3 

19.1 

19.9 

20.8 

21.6 

22.5 

23.4 

24.3 

25.2 

69 

16.9 

17.7 

18.5 

19.4 

20.2 

21.1 

21.9 

22.8 

23.7 

24.6 

25.6 

70 

17.2 

18.0 

18.8 

19.6 

20.5 

21.4 

22.3 

23.2 

24.1 

25.0 

25.9 

71 

17.4 

18.2 

19.1 

19.9 

20.8 

21.7 

22.6 

23.5 

24.4 

25.4 

26.3 

72 

17.7 

18.5 

19.3 

20.2 

21.1 

22.0 

22.9 

23.8 

24.8 

25.7 

26.7 

73 

17.9 

18.8 

19.6 

20.5 

21.4 

22.3 

23.2 

24.2 

25.1 

26.1 

27.1 

74 

18.2 

19.0 

19.9 

20.8 

21.7 

22.6 

23.5 

24.5 

25.5 

26.4 

27.4 

75 

18.4 

19.3 

20.1 

21.0 

22.0 

22.9 

23.9 

24.8 

25.8 

26.8 

27.8 

76 

18.6 

19.5 

20.4 

21.3 

22.3 

23.2 

24.2 

25.1 

26.1 

27.1 

28.2 

77 

18.9 

19.8 

20.7 

21.6 

22.6 

23.5 

24.5 

25.5 

26.5 

27.5 

28.5 

78 

19.1 

20.0 

21.0 

21.9 

22.8 

23.8 

24.8 

25.8 

26.8 

27.9 

28.9 

79 

19.4 

20.3 

21.2 

22.2 

23.1 

24.1 

25.1 

26.1 

27.2 

28.2 

29.3 

80 

19.6 

20.5 

21.5 

22.5 

23.4 

24.4 

25.4 

26.5 

27.5 

28.6 

29.7 

81 

19.9 

20.8 

21.8 

22.7 

23.7 

24.7 

25.8 

26.8 

27.9 

28.9 

30.0 

82 

20.1 

21.1 

22.0 

23.0 

24.0 

25.0 

26.1 

27.1 

28.2 

29.3 

30.4 

83 

20.4 

21.3 

22.3 

23.3 

24.3 

25.3 

26.4 

27.5 

28.5 

29.6 

30.8 

84 

20.6 

21.6 

22.6 

23.6 

24.6 

25.6 

26.7 

27.8 

28.9 

30.0 

31.1 

85 

20.8 

21.8 

22.8 

23.9 

24.9 

26.0 

27.0 

28.1 

29.2 

30.4 

31.5 

86 

21.1 

22.1 

23.1 

24.1 

25.2 

26.3 

27.3 

28.5 

29.6 

30.7 

31.9 

87 

21.3 

22.3 

23.4 

24.4 

25.5 

26.6 

27.7 

28.8 

29.9 

31.1 

32.2 

88 

21.6 

22.6 

23.6 

24.7 

25.8 

26.9 

28.0 

29.1 

30.3 

31.4 

32.6 

89 

21.8 

22.9 

23.9 

25.0 

26.1 

27.2 

28.3 

29.4 

30.6 

31.8 

33.0 

90 

22.1 

23.1 

24.2 

25.3 

26.4 

27.5 

28.6 

29.8 

31.0 

32.1 

33.4 

91 

22.3 

23.4 

24.4 

25.5 

26.7 

27.8 

28.9 

30.1 

31.3 

32.5 

33.7 

92 

22.6 

23.6 

24.7 

25.8 

26.9 

28.1 

29.3 

30.4 

31.6 

32.9 

34.r 

93 

22.8 

23.9 

25.0 

26.1 

27.2 

28.4 

29.6 

30.8 

32.0 

33.2 

34.5 

94 

23.1 

24.1 

25.3 

26.4 

27.5 

28.7 

29.9 

31.1 

32.3 

33.6 

34.8 

95 

23.3 

24.4 

25.5 

26.7 

27.8 

29.0 

30.2 

31.4 

32.7 

33.9 

35.2 

96 

23.5 

24.7 

25.8 

26.9 

28.1 

29.3 

30.5 

31.8 

33.0 

34.3 

35.6 

97 

23.8 

24.9 

26.1 

27.2 

28.4 

29.6 

30.8 

32.1 

33.4 

34.6 

36.0 

98 

24.0 

25.2 

26.3 

27.5 

28.7 

29.9 

31.2 

32.4 

33.7 

35.0 

36.3 

99 

24.3 

25.4 

26.6 

27.8 

29.0 

30.2 

31.5 

32.8 

34.1 

35.4 

36.7 

100 

24.5 

25.7 

26.9 

28.1 

29.3 

30.5 

31.8 

33.1 

34.4 

35.7 

37.1 

600 

147.2 

154.1 

161.2 

168.4 

175.7 

183.2 

190.8 

198.5 

206.4 

214.3 

222.4 

700 

171.7 

179.8 

188.1 

196.5 

205.0 

213.7 

222.6 

231.6 

240.8 

250.0 

259.4 

800 

196.1 

205.4 

214.9 

224.6 

234.3 

244.2 

254.4 

264.7 

275.2 

285.8 

296.5 

900 

220.8 

231.2 

241.8 

252.7 

263.7 

274.8 

286.2 

297.8 

309.7 

321.5 

333.7 

1.33 

1.35 

1.37 

1.39 

1.41 

1.44 
Factor 

1.47 

1.50 

1.52 

1.56 

1.59 

To  Change  Dep.  into  Long.  Diff.  Multiply  Tabular  Number  by 
Factor  at  Foot  of  Column  and  add  Product  to  Dep. 


176 


Table  2 


To  Change  Long.  Diff.  into  Dep.,  subtract  Tabular  Number 

FROM  Long.  Diff. 


Long. 
Diff. 

OR 

Dep. 

Middle  Latitude 

52° 

53° 

54° 

55° 

56° 

57° 

58° 

59° 

60° 

1 

0.4 

0.4 

0.4 

0.4 

0.4 

0.5 

0.5 

0.5 

0.5 

2 

0.8 

0.8 

0.8 

0.9 

0.9 

0.9 

0.9 

1.0 

1.0 

3 

1.2 

1.2 

1.2 

1.3 

1.3 

1.4 

1.4 

1.5 

1.5 

4 

1.5 

1.6 

1.6 

1.7 

1.8 

1.8 

1.9 

1.9 

2.0 

5 

1.9 

2.0 

2.1 

2.1 

2.2 

2.3 

2.4 

2.4 

2.5 

6 

2.3 

2.4 

2.5 

2.6 

2.6 

2.7 

2.8 

2.9 

3.0 

7 

2.7 

2.8 

2.9 

3.0 

3.1 

3.2 

3.3 

3.4 

3.5 

8 

3.1 

3.2 

3.3 

3.4 

3.5 

3.6 

3.8 

3.9 

4.0 

9 

3.5 

3.6 

3.7 

3.8 

4.0 

4.1 

4.2 

4.4 

4.5 

10 

3.8 

4.0 

4.1 

4.3 

4.4 

4.6 

4.7 

4.8 

5.0 

11 

4.2 

4.4 

4.5 

4.7 

4.8 

5.0 

5.2 

5.3 

5.5 

12 

4.6 

4.8 

4.9 

5.1 

5.3 

5.5 

5.6 

5.8 

6.0 

13 

5.0 

5.2 

5.4 

5.5 

5.7 

5.9 

6.1 

6.3 

6.5 

14 

5.4 

5.6 

5.8 

6.0 

6.2 

6.4 

6.6 

6.8 

7.0 

15 

5.8 

6.0 

6.2 

6.4 

6.6 

6.8 

7.1 

7.3 

7.5 

16 

6.1 

6.4 

6.6 

6.8 

7.1 

7.3 

7.5 

7.8 

8.0 

17 

6.5 

6.8 

7.0 

7.2 

7.5 

7.7 

8.0 

8.2 

8.5 

18 

6.9 

7.2 

7.4 

7.7 

7.9 

8.2 

8.5 

8.7 

9.0 

19 

7.3 

7.6 

7.8 

8.1 

8.4 

8.7 

8.9 

9.2 

9.5 

20 

7.7 

8.0 

8.2 

8.5 

8.8 

9.1 

9.4 

9.7 

10.0 

21 

8.1 

8.4 

8.7 

9.0 

9.3 

9.6 

9.9 

10.2 

10.5 

22 

8.5 

8.8 

9.1 

9.4 

9.7 

10.0 

10.3 

10.7 

11.0 

23 

8.8 

9.2 

9.5 

9.8 

10.1 

10.5 

10.8 

11.2 

11.5 

24 

9.2 

9.6 

9.9 

10.2 

10.6 

10.9 

11.3 

11.6 

12.0 

25 

9.6 

10.0 

10.3 

10.7 

11.0 

11.4 

11.8 

12.1 

12.5 

26 

10.0 

10.4 

10.7 

11.1 

11.5 

11.8 

12.2 

12.6 

13.0 

27 

10.4 

10.8 

11.1 

11.5 

11.9 

12.3 

12.7 

13.1 

13.5 

28 

10.8 

11.1 

11.5 

11.9 

12.3 

12.8 

13.2 

13.6 

14.0 

29 

11.1 

11.5 

12.0 

12.4 

12.8 

K  13.2 

13.6 

14.1 

14.5 

30 

11.5 

11.9 

12.4 

12.8 

13.2 

13.7 

14.1 

14.5 

15.0 

31 

11.9 

12.3 

12.8 

13.2 

13.7 

14.1 

14.6 

15.0 

15.5 

32 

12.3 

12.7 

13.2 

13.6 

14.1 

14.6 

15.0 

15.5 

16.0 

33 

12.7 

13.1 

13.6 

14.1 

14.5 

15.0 

15.5 

16.0 

16.5 

34 

13.1 

13.5 

14.0 

14.5 

15.0 

15.5 

16.0 

16.5 

17.0 

35 

13.5 

13.9 

14.4 

14.9 

15.4 

15.9 

16.5 

17.0 

17.5 

36 

13.8 

14.3 

14.8 

15.4 

15.9 

16.4 

16.9 

17.5 

18.0 

37 

14.2 

14.7 

15.3 

15.8 

16.3 

16.8 

17.4 

17.9 

18.5 

38 

14.6 

15.1 

15.7 

16.2 

16.8 

17.3 

17.9 

18.4 

19.0 

39 

15.0 

15.5 

16.1 

16.6 

17.2 

17.8 

18.3 

18.9 

19.5 

40 

15.4 

15.9 

16.5 

17.1 

17.6 

18.2 

18.8 

19.4 

20.0 

41 

15.8 

16.3 

16.9 

17.5 

18.1 

18.7 

19.3 

19.9 

20.5 

42 

16.1 

16.7 

17.3 

17.9 

18.5 

19.1 

19.7 

20.4 

21.0 

43 

16.5 

17.1 

17.7 

18.3 

19.0 

19.6 

20.2 

20.9 

21.5 

44 

16.9 

17.5 

18.1 

i8.8 

19.4 

20.0 

20.7 

21.3 

22.0 

45 

17.3 

17.9 

18.5 

19.2 

19.8 

20.5 

21.2 

21.8 

22.5 

46 

17.7 

18.3 

19.0 

19.6 

20.3 

20.9 

21.6 

22.3 

23.0 

47 

18.1 

18.7 

19.4 

20.0 

20.7 

21.4 

22.1 

22.8 

23.5 

48 

18.4 

19.1 

19.8 

20.5 

21.2 

21.9 

22.6 

23.3 

24.0 

49 

18.8 

19.5 

20.2 

20.9 

21.6 

22.3 

23.0 

23.8 

24.5 

50 

19.2 

19.9 

20.6 

21.3 

22.0 

22.8 

23.5 

24.2 

25.0 

100 

38.4 

39.8 

41.2 

42.6 

44.1 

45.5 

47.0 

48.5 

50.0 

200 

76.9 

79.6 

82.4 

85.3 

88.2 

91.1 

94.0 

97.0 

100.0 

300 

115.3 

119.5 

123.7 

127.9 

132.2 

136.6 

141.0 

145.5 

150.0 

400 

153.7 

159.3 

164.9 

170.6 

176.3 

182.2 

188.1 

194.0 

200.0 

500 

192.2 

199.1 

206.1 

213.2 

220.4 

227.7 

235.0 

242.5 

250.0 

1.62 

1.66 

1.70 

1.74 

1.79 
Factor 

1.84 

1.89 

1.94 

2.00 

To  Change  Dep.  into  Long.  Diff.,  Multiply  Tabular  Number  by 
Factor  at  Foot  of  Column  and  add  Product  to  Dep. 


Table  2 


111 


To  Change  Long.  Diff.  into  Dep.  SUBTRACT  Tabular  Number 

FROM  Long.  Diff. 


Long. 
Diff. 

OR 

Dep. 

Middle  Latitude 

52° 

53° 

54° 

55° 

56° 

57° 

58° 

59° 

60° 

51 

~i9^(r 

20.3 

21.0 

21.7 

22.5 

23.2 

24.0 

24.7 

25.5 

62 

20.0 

20.7 

21.4 

22.2 

22.9 

23.7 

24.4 

25.2 

26.0 

53 

20.4 

21.1 

21.8 

22.6 

23.4 

24.1 

24.9 

25.7 

26.5 

54 

20.8 

21.5 

22.3 

23.0 

23.8 

24.6 

25.4 

26.2 

27.0 

55 

21.1 

21.9 

22.7 

23.5 

24.2 

25.0 

25.9 

26.7 

27.5 

56 

21.5 

22.3 

23.1 

23.9 

24.7 

25.5 

26.3 

27.2 

28.0 

57 

21.9 

22.7 

23.5 

24.3 

25.1 

26.0 

26.8 

27.6 

28.5 

58 

22.3 

23.1 

23.9 

24.7 

25.6 

26.4 

27.3 

28.1 

29.0 

69 

22.7 

23.5 

24.3 

25.2 

26.0 

26.9 

27.7 

28.6 

29.5 

60 

23.1 

23.9 

24.7 

25.6 

26.4 

27.3 

28.2 

29.1 

30.0 

61 

23.4 

24.3 

25.1 

26.0 

26.9 

27.8 

28.7 

29.6 

30.5 

62 

23.8 

24.7 

25.6 

26.4 

27.3 

28.2 

29.1 

30.1 

31.0 

63 

24.2 

25.1 

26.0 

26.9 

27.8 

28.7 

29.6 

30.6 

31.5 

64 

24.6 

25.5 

26.4 

27.3 

28.2 

29.1 

30.1 

31.0 

32.0 

65 

25.0 

25.9 

26.8 

27.7 

28.7 

29.6 

30.6 

31.5 

32.5 

66 

25.4 

26.3 

27.2 

28.1 

29.1 

30.1 

31.0 

32.0 

33.0 

67 

25.8 

26.7 

27.6 

28.6 

29.5 

30.5 

31.5 

32.5 

33.5 

68 

26.1 

27.1 

28.0 

29.0 

30.0 

31.0 

32.0 

33.0 

34.0 

69 

26.5 

27.5 

28.4 

29.4 

30.4 

31.4 

32.4 

33.5 

34.5 

70 

26.9 

27.9 

28.9 

29.8 

30.9 

31.9 

32.9 

33.9 

35.0 

71 

27.3 

28.3 

29.3 

30.3 

31.3 

32.3 

33.4 

34.4 

35.5 

72 

27.7 

28.7 

29.7 

30.7 

31.7 

32.8 

33.8 

34.9 

36.0 

73 

28.1 

29.1 

30.1 

31.1 

32.2 

33.2 

34.3 

35.4 

36.5 

74 

28.4 

29.5 

30.5 

31.6 

32.6 

33.7 

34.8 

35.9 

37.0 

75 

28.8 

29.9 

30.9 

32.0 

33.1 

34.2 

35.3 

36.4 

37.5 

76 

29.2 

30.3 

31.3 

32.4 

33.5 

34.6 

35.7 

36.9 

38.0 

77 

29.6 

30.7 

31.7 

32.8 

33.9 

35.1 

36.2 

37.3 

38.5 

78 

30.0 

31.1 

32.2 

33.3 

34.4 

35.5 

36.7 

37.8 

39.0 

79 

30.4 

31.5 

32.6 

33.7 

34.8 

36.0 

37.1 

38.3 

39.5 

80 

30.7 

31.9 

33.0 

34.1 

35.3 

36.4 

37.6 

38.8 

40.0 

81 

31.1 

32.3 

33.4 

34.5 

35.7 

36.9 

38.1 

39.3 

40.5 

82 

31.5 

32.7 

33.8 

35.0 

36.1 

37.3 

38.5 

39.8 

41.0 

83 

31.9 

33.0 

34.2 

35.4 

36.6 

37.8 

39.0 

40.3 

41.5 

84 

32.3 

33.4 

34.6 

35.8 

37.0 

38.3 

39.5 

40.7 

42.0 

85 

32.7 

33.8 

35.0 

36.2 

37.5 

38.7 

40.0 

41.2 

42.5 

86 

33.1 

34.2 

35.5 

36.7 

37.9 

39.2 

40.4 

41.7 

43.0 

87 

33.4 

34.6 

35.9 

37.1 

38.4 

39.6 

40.9 

42.2 

43.5 

88 

33.8 

35.0 

36.3 

37.5 

38.8 

40.1 

41.4 

42.7 

44.0 

89 

34.2 

35.4 

36.7 

38.0 

39.2 

40.5 

41.8 

43.2 

44.5 

90 

34.6 

35.8 

37.1 

38.4 

39.7 

41.0 

42.3 

43.6 

45.0 

91 

35.0 

36.2 

37.5 

38.8 

40.1 

41.4 

42.8 

44.1 

45.5 

92 

35.4 

36.6 

37.9 

39.2 

40.6 

41.9 

43.2 

44.6 

46.0 

93 

35.7 

37.0 

38.3 

39.7 

41.0 

42.3 

43.7 

45.1 

46.5 

94 

36.1 

37.4 

38.7 

40.1 

41.4 

42.8 

44.2 

45.6 

47.0 

95 

36.5 

37.8 

39.2 

40.5 

41.9 

43.3 

44.7 

46.1 

47.5 

96 

36.9 

38.2 

39.6 

40.9 

42.3 

43.7 

45.1 

46.6 

48.0 

97 

37.3 

38.6 

40.0 

41.4 

42.8 

44.2 

45.6 

47.0 

48.5 

98 

37.7 

39.0 

40.4 

41.8 

43.2 

44.6 

46.1 

47.5 

49.0 

99 

38.0 

39.4 

40.8 

42.2 

43.6 

45.1 

46.5 

48.0 

49.5 

100 

38.4 

39,8 

41.2 

42.6 

44.1 

45.5 

47.0 

48.5 

50.0 

600 

230.6 

238.9 

247.3 

255.9 

264.5 

273.2 

282.0 

291.0 

300.0 

700 

269.2 

279.7 

288.6 

298.5 

308.6 

318.7 

329.0 

339.6 

350.0 

800 

307.5 

319.5 

329.8 

341.2 

352.6 

364.3 

376.1 

388.0 

400.0 

900 

346.0 

358.3 

371.1 

383.8 

396.8 

409.9 

423.2 

436.6 

450.0 

1.63 

1.66 

1.70 

1.74 

F 

1.79 

ACTOR 

1.84 

1.89 

1.94 

2.00 

To  Change  Dep.  into  Long.  Diff.  Multiply  Tabular  Number  by 
Factor  at  Foot  of  Column  and  add  Product  to  Dep. 


178 


Table  3.    Number  Logarithms 


100 

01 
02 
03 

04 
05 
06 

07 
08 
09 

110 

11 
12 
13 

14 
15 
16 

17 

18 
19 

120 


21 

22 
23 

24 
25 
26 

27 
28 
29 


130 


31 
32 
33 

34 
35 
36 

37 
38 
39 


140 

41 
42 
43 

44 
45 
46 

47 
48 
49 

150 


00  000 


432 

860 

01284 

703 

02119 

531 

938 

03  342 

743 


04139 

532 

922 

05  308 

6<)0 

06070 

446 

819 

07188 
555 


918 


08279 
636 
991 

09342 

691 

10037 

380 

721 

11059 


394 


727 

12  057 

385 

710 

13033 

354 

672 

988 

14  301 


613 


922 

15229 

534 

836 

16 137 

435 

732 

17026 

319 

609 


043 


475 
903 
326 

745 
](S0 
572 

979 
383 

782 


179 


571 
961 
346 

729 
108 
483 

856 
225 
591 


954 


314 

672 
*026 

377 
726 
072 

415 

755 
093 


428 


760 
090 
418 

743 

066 
386 

704 

*019 

333 


644 


953 
259 
564 

866 
167 
465 

761 
056 
348 

638 


087 


518 
945 
368 

787 
202 
612 

*019 
423 

822 


218 


610 

999 
385 

767 
145 
521 

893 
262 

628 


990 


*027 


350 

707 

*061 

412 
760 
106 

449 

789 
126 


461 


793 
123 
450 

775 

098 
418 

735 

*051 

364 


675 


983 
290 
594 

897 
197 
495 

791 

085 
377 

667 


130 


561 
988 
410 

828 
243 
653 

*060 
4(33 

862 


258 


650 

*038 

423 

805 
183 
558 

930 
298 
6()4 


*063 


386 

743 

*096 

447 
795 
140 

483 
823 
160 


494 


826 
156 
483 

808 
130 
450 

767 

*082 

395 


706 


*014 
320 
625 

927 

227 
524 

820 
114 
406 

696 


173 


604 

*030 

452 

870 
284 
694 

*100 
503 
902 


297 


689 

*077 

461 

843 
221 
595 

967 
335 
700 


422 

778 

*132 

482 
830 
175 

517 

857 
193 


528 


860 
189 
516 

840 
162 
481 

799 

*114 

426 


737 


*045 
351 
655 

957 
256 
554 

850 
143 
435 

725 


217 


647 

*072 

494 

912 
325 

735 

*141 
543 
941 


33(5 


727 

*115 

500 

881 
258 
633 

*004 
372 

737 


*099 


458 

814 

*167 

517 
864 
209 

551 

890 
227 


561 


893 
222 
548 

872 
194 
513 

830 
*145 

457 


768 


*076 
381 
685 

987 
286 
584 

879 
173 
464 

754 


260 


689 

*115 

536 

953 
3()6 
776 

*181 
583 
981 


376 


766 

*154 

538 

918 
296 
670 

*041 

408 
773 


*135 


493 

849 
*202 

552 
899 
243 

585 
924 
261 


594 


926 

254 

581 

905 
226 
545 

862 

*176 

489 


799 


*106 
412 
715 

*017 
316 
613 

909 
202 
493 

782 


303 


732 

*157 

578 

995 
407 
816 

*222 

623 

*021 


415 


805 

*192 

576 

956 
333 

707 

*078 
445 
809 


^171 


529 

884 

*237 

587 
934 
278 

619 

958 
294 


628 


959 

287 
613 

937 

258 
577 

893 

*208 

520 


829 


*137 
442 
746 

*047 
346 
643 

938 
231 
522 

811 


346 


775 

*199 

620 

*036 
449 
857 

*262 

663 

*060 


454 


844 

*231 

614 

994 
371 

744 

*115 

482 
846 


*207 


565 

920 

*272 

621 

968 
312 

653 

992 
327 


661 


992 
320 
646 

969 
290 
609 

925 

*239 

551 


860 


*168 
473 
776 

*077 
376 
673 

967 
260 
551 

840 


389 


817 

*242 

662 

*078 
4^)0 
898 

*302 

703 

*100 


493 


883 

*269 

652 

*032 
408 
781 

*151 
518 

882 


*243 


600 

955 

*307 

656 

*003 

346 

687 

*025 

361 


694 


*024 
352 
678 

*001 

322 
640 

956 

*270 

582 


891 


*198 
503 
806 

*107 
406 
702 

997 
289 
580 

869 


Prop.  Pts. 


44 

43 

4.4 

4.3 

8.8 

8.6 

13.2 

12.9 

17.6 

17.2 

22.0 

21.5 

26.4 

25.8 

30.8 

30.1 

35.2 

34.4 

39.6 

38.7 

41 

40 

1 

4.1 

4.0 

2 

8.2 

8.0 

3 

12.3 

12.0 

4 

16.4 

16.0 

5 

20.5 

20.0 

() 

24.6 

24.0 

7 

28.7 

28.0 

8 

32.8 

32.0 

9 

36.9 

36.0 

38 

37 

3.8 

3.7 

7.6 

7.4 

11.4 

11.1 

15.2 

14.8 

19.0 

18.5 

22.8 

22.2 

26.6 

25.9 

30.4 

29.6 

34.2 

33.3 

35 

34 

3.5 

3.4 

7.0 

6.8 

10.5 

10.2 

14.0 

13.6 

17.5 

17.0 

21.0 

20.4 

24.5 

23.8 

28.0 

27.2 

31.5 

30.6 

32 

31 

3.2 

3.1 

6.4 

6.2 

9.6 

9.3 

12.8 

12.4 

16.0 

15.5 

19.2 

18.6 

22.4 

21.7 

25.6 

24.8 

28.8 

27.9 

42 

4.2 
8.4 
12.6 
16.8 
21.0 
25.2 
29.4 
33.6 
37.8 


39 

3.9 

7.8 
11.7 
15.6 
19.5 
23.4 
27.3 
31.2 
35.1 


36 

3.6 
7.2 
10.8 
14.4 
18.0 
21.6 
25.2 
28.8 
32.4 


33 

3.3 
6.6 
9.9 
13.2 
16.5 
19.8 
23.1 
26.4 
29.7 


30 

3.0 
6.0 
9.0 
12.0 
15.0 
18.0 
21.0 
24.0 
27.0 


Prop.  Pts. 


Table  3.    Number  Logarithms 


179 


0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Prop.  Pts. 

150 

17  609 

638 

667 

696 

725 

754 

782 

811 

840 

869 

51 
52 
53 

54 
55 
56 

57 
58 
59 

160 

898 

18184 

469 

752 

19033 

312 

590 

866 

20140 

926 
213 

498 

780 
061 
340 

618 
893 
167 

955 
241 
526 

808 
089 
368 

645 
921 
194 

984 
270 
554 

837 
117 
396 

673 
948 

222 

*013 
298 

583 

865 
145 
424 

700 
976 
249 

*041 
327 
611 

893 
173 
451 

728 

*003 

276 

*070 
355 
639 

921 
201 
479 

756 

*030 

303 

*099 
384 
667 

949 
229 
507 

783 

*058 

330 

*127 
412 
696 

977 
257 
535 

811 

*085 
358 

*156 
441 
724 

*005 
285 
562 

838 
*112 

385 

412 

439 

466 

493 

520 

548 

575 

602 

629 

656 

61 
62 
63 

64 
65 

67 
68 
69 

683 

952 

21219 

484 

748 

22  011 

272 
531 

789 

710 
978 
245 

511 

775 
037 

298 
557 
814 

737 
*005 

272 

537 
801 
063 

324 

583 
840 

763 

*032 
299 

564 
827 
089 

350 

608 
866 

7i)0 

•*059 

325 

590 
854 
115 

376 
634 
891 

817 

*085 

352 

617 
880 
141 

401 
660 
917 

844 

*H2 

378 

643 
906 
167 

427 

686 
943 

871 

*139 

405 

669 
932 
194 

453 
712 

968 

898 

*165 

431 

696 

958 
220 

479 

737 
994 

925 

*192 

458 

722 
985 
246 

505 

763 

*019 

1 
2 
3 
4 
5 
6 
7 
8 
9 

29 

2.9 
5.8 
8.7 
11.6 
14.5 
17.4 
20.3 
23.2 
26.1 

28 

2.8 
5.6 
8.4 
11.2 
14.0 
16.8 
19.6 
22.4 
25.2 

27 

2.7 
5.4 
8.1 
10.8 
13.5 
16.2 
18.9 
21.6 
24.3 

170 

23045 

070 

096 

121 

147 

172 

198 

223 

249 

274 

71 
72 
73 

74 
75 
76 

77 
78 
79 

300 
553 
805 

24  055 
304 
551 

797 

25  042 

285 

325 
578 
830 

080 
329 
576 

822 
066 
310 

350 
603 
855 

105 
353 
601 

846 
091 
334 

376 
629 
880 

130 
378 
625 

871 
115 

358 

401 
654 
905 

155 

403 
650 

895 
139 

382 

426 
679 
930 

180 
428 
674 

920 
164 

406 

452 
704 
955 

204 
452 
699 

944 

188 
431 

477 
729 
980 

229 

477 
724 

969 
212 
455 

502 

754 

*005 

254 
502 
748 

993 

237 
479 

528 

779 

*030 

279 
527 
773 

*018 
261 
503 

1 
2 
3 
4 
5 
6 
7 
8 
9 

26 

2.6 
5.2 
7.8 
10.4 
13.0 
15.6 
18.2 
20.8 
23.4 

25 

2.5 
5.0 
7.5 
10.0 
12.5 
15.0 
17.5 
20.0 
22.5 

24 

2.4 

4.8 

7.2 

9.6 

12.0 

14.4 

16.8 

19.2 

21.6 

180 

527 

551 

575 

600 

624 

648 

672 

696 

720 

744 

81 

82 
83 

84 
85 
86 

87 
88 
89 

768 

26007 

245 

482 
717 
951 

27184 
416 
646 

792 
031 
269 

505 
741 
975 

207 
439 
669 

816 
055 
293 

529 
764 
998 

231 

462 
692 

840 
079 
316 

553 

788 
*021 

254 

485 
715 

864 
102 
340 

576 

811 

*045 

277 
508 
738 

888 
126 
364 

600 

834 

*068 

300 
531 
761 

912 
150 

387 

623 

858 
*091 

323 
554 

784 

935 
174 
411 

647 

881 

*114 

346 

577 
807 

959 

198 
435 

670 

905 

*138 

370 

600 
830 

983 
221 
458 

694 

928 
*161 

393 
623 

852 

1 
2 
3 
4 
5 
6 
7 
8 
9 

23 

2.3 

4.6 

6.9 

9.2 

11.5 

13.8 

16.1 

18.4 

20.7 

22 

2.2 

4.4 

6.6 

8.8 

11.0 

13.2 

15.4 

17.6 

19.8 

21 

2.1 

4.2 
6.3 
8.4 
10.5 
12.6 
14.7 
16.8 
18.9 

190 

875 

898 

921 

944 

967 

989 

*012 

*035 

*058 

*081 

91 
92 
93 

94 
95 
96 

97 

98 
99 

28103 
330 
556 

780 

29003 

226 

447 

667 
885 

126 
353 

578 

803 
026 

248 

469 

688 
907 

149 
375 
601 

825 
048 
270 

491 
710 

929 

171 

398 
623 

847 
070 
292 

513 

732 
951 

194 
421 

646 

870 
092 
314 

535 
754 
973 

217 
443 

668 

892 
115 
336 

557 
776 
994 

240 
466 
691 

914 
137 
358 

579 

798 
*016 

262 
488 
713 

937 
159 
380 

601 

820 
*038 

285 
511 
735 

959 
181 
403 

623 

842 
*060 

307 
533 
758 

981 
203 
425 

645 

863 
*081 

200 

30103 

125 

146 

168 

190 

211 

233 

255 

276 

298 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Prop.  Pts. 

180 


Table  3.    Number  Logarithms 


0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Prop.  Pts. 

200 

30103 

125 

146 

168 

190 

211 

233 

255 

276 

298 

01 
02 
03 

320 
535 
750 

341 
557 
771 

363 
578 
792 

384 
600 
814 

406 
621 
835 

428 
643 
856 

449 

6<)4 
878 

471 

685 
899 

492 
707 
920 

514 

728 
942 

04 
05 
OG 

963 
31175 

387 

984 
197 
408 

*006 
218 
429 

*027 
239 
450 

*048 
260 
471 

*m9 

281 
492 

*091 
302 
513 

*112 
323 
534 

*133 
345 
555 

*154 
366 
576 

07 
08 
09 

597 

806 

32  015 

618 
827 
035 

639 

848 
056 

660 

869 
077 

681 
8^K) 
098 

702 
911 
118 

723 
931 
139 

744 
952 
160 

765 
973 
181 

785 
9i)4 
201 

210 

222 

243 

263 

284 

305 

325 

346 

366 

387 

408 

11 
12 
13 

428 
634 
838 

449 
(;54 
858 

469 
675 

879 

490 
695 
899 

510 
715 
919 

531 
736 
940 

552 
756 
960 

572 

777 
980 

593 

797 

*001 

613 

818 

*021 

1 
2 
3 
4 
5 
6 

}6}6 

2.2 
4.4 

6.6 

8.8 

11.0 

13.2 

2.1 
4.2 
6.3 
8.4 
10.5 
12.6 

}6yj 

2.0 
4.0 
6.0 
8.0 
10.0 
12.0 

14 
15 
16 

33041 
244 
445 

062 
264 
465 

082 
284 
486 

102 
3(M 
506 

122 
325 
526 

143 
345 
546 

163 
3(35 
566 

183 
385 
586 

203 
405 
606 

224 
425 
626 

17 

18 
19 

646 

846 

34044 

666 

866 
064 

686 
885 
084 

706 
905 
104 

726 
925 
124 

746 
945 
143 

766 
965 
163 

786 
985 
183 

806 

*005 

203 

826 

*025 

223 

7 
8 
9 

15.4 
17.6 
19.8 

14.7 
16.8 
18.9 

14.0 
16.0 
18.0 

220 

242 

262 

282 

301 

321 

341 

361 

380 

400 

420 

21 

22 
23 

439 
635 
830 

459 
655 

850 

479 
674 

869 

498 
694 
889 

518 
713 
908 

537 
733 
928 

557 
753 
947 

577 
772 
967 

596 

792 
986 

616 

811 

*005 

24 
25 
26 

35025 

218 
411 

044 
238 
430 

064 
257 
449 

083 
276 
468 

102 
295 

488 

122 

315 
507 

141 

334 
526 

160 
353 
545 

180 
372 
564 

199 
392 

583 

27 
28 
29 

603 

793 
984 

622 

813 

*003 

641 

832 

*021 

660 

851 

*040 

679 

870 

*059 

698 

889 

*078 

717 

908 

*097 

736 

927 

*116 

755 

946 

*135 

774 

965 

*154 

230 

36 173 

192 

211 

229 

248 

267 

286 

305 

324 

342 

31 
32 
33 

361 
549 
736 

380 

568 
754 

399 
586 
773 

418 
605 
791 

436 
624 

810 

455 
642 
829 

474 

mi 

847 

493 
680 
866 

511 

698 
884 

530 
717 
903 

1 
2 

3 
4 

5 

1.9 
3.8 
5.7 
7.6 
9.5 
11.4 

18 

1.8 
3.6 
5.4 
7.2 
9.0 
10.8 

17 

1.7 
3.4 
5.1 
6.8 
8.5 
10.2 

34 
35 
36 

922 

37107 
291 

&40 
125 
310 

959 
144 
328 

977 
162 
346 

996 
181 
365 

*014 
199 
383 

*033 
218 
401 

*051 
236 
420 

*070 
254 
438 

*088 
273 
457 

37 
38 
39 

475 

658 
840 

493 

676 
858 

511 
694 

876 

530 

712 
894 

548 
731 
912 

566 
749 
931 

585 

767 
949 

603 

785 
967 

621 
803 
985 

639 

822 
*003 

7 
8 
9 

13.3 
15.2 
17.1 

12.6 
14.4 
16.2 

11.9 
13.6 
15.3 

240 

38021 

039 

057 

075 

093 

112 

130 

148 

166 

184 

41 
42 
43 

202 
382 
561 

220 

399 
578 

238 
417 
596 

256 
435 
614 

274 

453 
632 

292 
471 
650 

310 
489 
668 

328 
507 
686 

at6 
525 
703 

364 
543 
721 

44 
45 
46 

739 

917 

39094 

757 
934 
111 

775 
952 
129 

792 
970 
146 

810 
987 
164 

828 

*005 

182 

846 

*023 

199 

863 

*041 

217 

881 

*058 

235 

899 

*076 

252 

47 

48 
49 

270 
445 
620 

287 
463 
637 

305 
480 
655 

322 
498 
672 

340 
515 
690 

358 
533 

707 

375 
550 
724 

393 
568 
742 

410 

585 
759 

428 
602 

777 

250 

794 

811 

829 

846 

863 

881 

898 

915 

933 

950 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Prop.  Pts. 

Table  3. 

Number  Logarithms 

181 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Prop.  Pts. 

250 

39  794 

811 

829 

846 

863 

881 

898 

915 

933 

950 

51 

967 

985 

*002 

*019 

*037 

*054 

*071 

*088 

*106 

*123 

52 

40140 

157 

175 

192 

209 

226 

243 

2()1 

278 

295 

53 

312 

329 

346 

364 

381 

398 

415 

432 

449 

466 

54 

483 

500 

518 

535 

552 

569 

586 

603 

620 

637 

55 

(554 

671 

688 

705 

722 

739 

756 

773 

790 

807 

56 

824 

841 

858 

875 

892 

909 

926 

943 

960 

976 

57 

993 

*010 

*027 

*044 

*061 

*078 

*095 

*ni 

*128 

*145 

58 

41 162 

179 

196 

212 

229 

246 

263 

280 

296 

313 

59 

330 

347 

363 

380 

397 

414 

430 

447 

464 

481 

260 

497 

514 

531 

547 

5()4 

581 

597 

614 

631 

647 

61 

664 

681 

697 

714 

731 

747 

764 

780 

797 

814 

18 

17 

16 

62 

8;30 

847 

863 

880 

896 

913 

929 

946 

963 

979 

1 

1.8 

1.7 

1.6 

63 

996 

*012 

*029 

*045 

*062 

*078 

*095 

*111 

*127 

*144 

2 

3.6 

3.4 

3.2 

64 

42  160 

177 

193 

210 

226 

243 

259 

275 

292 

308 

3 
4 

5 
6 

5.4 

7.2 

9.0 

10.8 

5.1 

6.8 

8.5 

10.2 

4.8 
6.4 
8.0 
9.6 

65 

325 

341 

357 

374 

390 

406 

423 

439 

455 

472 

m 

488 

504 

521 

537 

553 

570 

586 

602 

619 

635 

67 

651 

667 

684 

700 

716 

732 

749 

765 

781 

797 

7 

12.6 

11.9 

U.2 

68 

813 

830 

846 

862 

878 

894 

911 

927 

943 

959 

8 

14.4 

13.6 

12.8 

69 

975 

991 

*008 

*024 

*040 

*056 

*072 

*088 

*104 

*120 

9 

16.2 

15.3 

14.4 

270 

43136 

152 

169 

185 

201 

217 

233 

249 

265 

281 

71 

297 

313 

329 

345 

361 

377 

393 

409 

425 

441 

72 

457 

473 

489 

505 

521 

537 

553 

569 

584 

600 

73 

616 

632 

648 

664 

680 

696 

712 

727 

743 

759 

74 

775 

791 

807 

823 

838 

854 

870 

886 

f)02 

917 

75 

933 

949 

965 

981 

996 

*012 

*028 

*044 

*059 

*075 

76 

44  091 

107 

122 

138 

154 

170 

185 

201 

217 

232 

77 

248 

264 

279 

295 

311 

326 

342 

358 

373 

389 

78 

404 

420 

436 

451 

467 

483 

498 

514 

529 

545 

79 

5()0 

576 

592 

607 

623 

638 

654 

669 

685 

700 

280 

716 

731 

747 

762 

778 

793 

809 

824 

840 

855 

81 

871 

886 

902 

917 

932 

948 

963 

979 

994 

*010 

15 

14 

82 

45025 

040 

056 

071 

086 

102 

117 

133 

148 

163 

1 
2 

1.5 
3.0 

1.4 

2.8 

83 

179 

194 

209 

225 

240 

255 

271 

286 

301 

317 

84 

332 

347 

362 

378 

393 

408 

423 

439 

454 

469 

3 

4.5 

4.2 

85 

484 

500 

515 

530 

545 

561 

576 

591 

606 

621 

4 

6.0 

5.6 

86 

637 

652 

667 

682 

697 

712 

728 

743 

758 

773 

5 
6 

7.5 
9.0 

7.0 
8.4 

8V 

788 

803 

818 

834 

849 

864 

879 

81^4 

909 

924 

7 

10.5 

9.8 

88 

939 

954 

969 

984 

*000 

*015 

*030 

*045 

■*(m 

*075 

8 

12.0 

11.2 

89 

46  090 

105 

120 

135 

150 

165 

180 

195 

210 

225 

9 

13.5 

12.6 

290 

240 

255 

270 

285 

300 

315 

330 

345 

359 

374 

91 

389 

404 

419 

434 

449 

464 

479 

494 

509 

523 

92 

538 

553 

568 

583 

598 

613 

627 

642 

657 

672 

93 

687 

702 

716 

731 

746 

761 

776 

790 

805 

820 

94 

835 

850 

864 

879 

894 

909 

923 

938 

953 

967 

95 

982 

997 

*012 

*026 

*041 

*056 

*070 

*085 

*100 

*114 

96 

47129 

144 

159 

173 

188 

202 

217 

232 

246 

261 

97 

276 

290 

305 

319 

334 

349 

363 

378 

392 

407 

98 

422 

436 

451 

4()5 

480 

494 

509 

524 

538 

553 

99 

567 

582 

596 

611 

625 

640 

654 

669 

683 

698 

300 

712 

727 

741 

756 

770 

784 

799 

813 

828 

842 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Prop.  Pts. 

182 


Table  3.    Number  Logarithms 


0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Prop.  Pts. 

300 

47  712 

727 

741 

756 

770 

7M 

799 

813 

828 

842 

01 
02 
03 

04 
05 
06 

07 

08 
09 

857 

48  001 

144 

287 
430 
572 

714 

855 
996 

871 
015 
159 

302 
444 
586 

728 

869 

*010 

885 
029 
173 

316 
458 
601 

742 

883 
*024 

900 
044 

187 

330 
473 
615 

756 

897 

*038 

914 

058 
202 

344 

487 
629 

770 
911 

*052 

929 
073 
216 

359 
501 
643 

785 

926 

*066 

943 

087 
230 

373 
515 
657 

799 
940 

*080 

958 
101 
244 

387 
530 
671 

813 
954 

*094 

972 
116 
259 

401 
544 

686 

827 

968 

*108 

986 
130 
273 

416 
558 
700 

841 

982 

*122 

310 

49136 

150 

164 

178 

192 

206 

220 

234 

248 

262 

14 

1.4 

2.8 
4.2 
5.6 
7.0 
8.4 
9.8 
11.2 
12.6 

11 
12 
13 

14 
15 
16 

17 
18 
19 

276 
415 
554 

693 
831 
969 

50106 
243 
379 

290 
429 
568 

707 
845 
982 

120 
256 
393 

304 
443 

582 

721 

859 
996 

133 
270 
406 

318 
457 
596 

734 

872 
*010 

147 

284 
420 

332 
471 
610 

748 

886 

*024 

161 

297 
433 

346 
485 
624 

762 

900 

*037 

174 

311 

447 

360 
499 
638 

776 

914 

*051 

188 
325 
461 

374 
513 
651 

790 

927 

*065 

202 

338 
474 

388 
527 
665 

803 

941 

*079 

215 
352 

488 

402 
541 
679 

817 

955 

*092 

229 
365 
501 

1 

2 
3 
4 
5 
6 
7 
8 
9 

1.5 

3.0 

4.5 

6.0 

7.5 

9.0 

10.5 

12.0 

13.5 

320 

515 

529 

542 

556 

569 

583 

596 

610 

623 

637 

21 

22 
23 

24 
25 
26 

27 
28 
29 

651 
786 
920 

51055 

188 
322 

455 

587 
720 

664 
799 
934 

068 
202 
335 

468 
601 
733 

678 
813 
947 

081 
215 
348 

481 
614 

746 

691 
826 
961 

095 
228 
362 

495 
627 
759 

705 
840 
974 

108 
242 
375 

508 
640 

772 

718 
853 

987 

121 

.255 

388 

521 
654 
786 

732 

866 

*001 

135 

268 
402 

534 

6(i7 
799 

745 

880 

*014 

148 
282 
415 

548 
680 
812 

759 
893 

*028 

162 
295 

428 

561 
693 

825 

772 

907 

*041 

175 
308 
441 

574 
706 
8:38 

330 

851 

865 

878 

891 

904 

917 

9-30 

943 

957 

970 

31 
32 
33 

34 
35 
36 

37 
38 
39 

983 

52114 

244 

375 
504 
6M 

763 

892 

53020 

996 
127 
257 

388 
517 
647 

776 
905 
033 

*009 
140 
270 

401 
530 
660 

789 
917 
046 

*022 
153 

284 

414 
543 

673 

802 
930 
058 

*035 
166 
297 

427 
556 
686 

815 
943 
071 

*048 
179 
310 

440 
569 
699 

827 
956 
084 

*061 
192 
323 

453 
582 
711 

840 
969 
097 

*075 
205 
336 

4m 

595 

724 

853 
982 
110 

*088 
218 
349 

479 
608 
737 

866 
994 
122 

*101 
231 
362 

492 
621 
750 

879 

*007 

135 

1 

2 
3 
4 
5 
6 
7 
8 
9 

13 

1.3 
2.6 
3.9 
5.2 
6.5 
7.8 
9.1 
10.4 
11.7 

12 

1.2 
2.4 
3.6 
4.8 
6.0 
7.2 
8.4 
9.6 
10.8 

340 

148 

161 

173 

186 

199 

212 

224 

237 

250 

263 

41 
42 
43 

44 
45 
46 

47 

48 
49 

275 
403 
529 

656 

782 
908 

54033 

158 
283 

288 
415 
542 

668 
794 
920 

045 
170 
295 

301 
428 
555 

681 
807 
933 

058 
183 
307 

314 
441 
567 

694 
820 
945 

070 
195 
320 

326 
453 
580 

706 
832 
958 

083 
208 
332 

339 
4<>6 
593 

719 
845 
970 

095 
220 
345 

352 
479 
605 

732 

857 
983 

108 
233 
357 

364 
491 
618 

744 
870 
995 

120 

245 
370 

377 
504 
631 

757 

882 

*008 

133 

258 
382 

390 
517 
643 

769 

895 

*020 

145 
270 
394 

350 

407 

419 

432 

444 

456 

469 

481 

4t)4 

506 

518 

0 

1 

2 

3 

4   5 

6 

7 

8 

9 

Prop.  Pts. 

Table  3.    Number  Logarithms 


183 


0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Prop.  Pts. 

350 

54  407 

419 

432 

444 

456 

469 

481 

494 

506 

518 

51 
52 
53 

54 
55 
56 

57 
58 
59 

531 
654 

777 

900 

55  023 

145 

267 

388 
509 

543 

667 
790 

913 
035 
157 

279 
400 
522 

555 
679 
802 

925 
047 
169 

291 
413 
534 

568 
691 
814 

937 
0(K) 
182 

303 
425 
546 

580 
704 
827 

949 
072 
194 

315 
437 

558 

593 
716 
839 

962 

084 
206 

328 
449 
570 

605 
728 
851 

974 
096 
218 

340 
461 

582 

617 
741 

864 

986 
108 
230 

352 
473 
594 

()30 
753 
876 

998 
121 
242 

364 

485 
606 

642 

765 
888 

*011 
133 
255 

376 

497 
618 

360 

630 

642 

654 

me 

678 

691 

703 

715 

727 

739 

61 
62 
63 

64 
65 
66 

67 

68 
69 

751 
871 
991 

56110 
229 
348 

467 

585 
703 

763 

883 
*003 

122 
241 
360 

478 
597 
714 

775 

895 

*015 

134 
253 
372 

490 
608 
726 

787 

907 

*027 

146 
265 

384 

502 
620 

738 

799 

919 

*038 

158 
277 
396 

514 

632 
750 

811 

931 

*050 

170 
289 
407 

526 
644 
761 

823 

943 

*062 

182 
301 
419 

538 
656 
773 

835 
955 

-*074 

194 
312 
431 

549 

667 

785 

847 

967 

*086 

205 
324 
443 

561 
679 
797 

859 
979 

*098 

217 
336 
455 

573 
691 

808 

1 
2 
3 
4 
5 
6 
7 
8 
9 

13 

1.3 
2.6 
3.9 
5.2 
6.5 
7.8 
9.1 
10.4 
11.7 

12 

1.2 

2.4 
3.6 

4.8 
6.0 
7.2 
8.4 
9.6 
10.8 

370 

820 

832 

844 

855 

867 

879 

891 

902 

914 

926 

71 
72 
73 

74 

75 
76 

77 
78 
79 

937 

57  054 

171 

287 
403 
519 

634 

749 
864 

949 
066 
183 

299 
415 
530 

646 
761 

875 

961 

078 
194 

310 
426 
542 

657 

772 
887 

972 
089 
206 

322 
438 
553 

669 
784 
898 

984 
101 
217 

334 
449 
565 

680 
795 
910 

9£K) 
113 
229 

345 
4<il 
576 

692 
807 
921 

*008 
124 
241 

357 
473 

588 

703 

818 
933 

*019 
136 
252 

368 
484 
600 

715 
830 
944 

*031 
148 
264 

380 
496 
611 

726 
841 
955 

*043 
159 
276 

392 

507 
623 

738 
852 
967 

380 

978 

990 

*001 

*013 

*024 

*035 

*047 

*058 

*070 

*081 

81 

82 
83 

84 
85 
86 

87 
88 
89 

58  092 
206 
320 

433 
546 
659 

771 
883 
995 

104 

218 
331 

444 

557 
670 

782 

894 

*006 

115 
229 
343 

456 
569 
681 

794 

906 

*017 

127 

240 
354 

467 
580 
692 

805 

917 

*028 

138 
252 
365 

478 
591 
704 

816 

928 

*040 

149 
263 
377 

490 
602 
715 

827 

939 

*051 

161 

274 
388 

501 
614 
726 

838 

950 

*062 

172 

286 
399 

512 

625 

737 

850 

961 

*073 

184 
297 
410 

524 
636 
749 

861 

973 

*084 

195 
309 
422 

535 

647 
760 

872 

984 

*095 

1 
2 
3 
4 
5 
6 
7 
8 
9 

11 

1.1 
2.2 
3.3 
4.4 
5.5 
6.6 
7.7 
8.8 
9.9 

10 

1.0 
2.0 
3.0 
4.0 
5.0 
6.0 
7.0 
8.0 
9.0 

390 

59  106 

118 

129 

140 

151 

162 

173 

184 

195 

207 

91 
92 
93 

94 
95 
96 

97 
98 
99 

218 
329 
439 

550 
660 

770 

879 

988 

60  097 

229 
340 
450 

561 
671 

780 

890 
999 
108 

240 
351 
461 

572 
682 
791 

901 

*010 

119 

251 
362 
472 

583 
693 

802 

912 

*021 

130 

262 
373 
483 

594 
704 
813 

923 

*032 

141 

273 
384 
494 

605 
715 
824 

934 

*043 

152 

284 
395 
506 

616 
726 

835 

945 

*054 

163 

295 
406 
517 

627 
737 
846 

956 

*065 

173 

306 
417 
528 

638 

748 
857 

966 

*076 

184 

318 
428 
539 

649 

759 
868 

977 

*086 

195 

400 

206 

217 

228 

239 

249 

260 

271 

282 

293 

304 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Prop.  Pts. 

184 


Table  3.    Number  Logarithms 


0 

1 

2 

3 

4 

6 

6 

7 

8 

9 

Prop.  Pts. 

400 

60  206 

217 

228 

239 

249 

260 

271 

282 

293 

304 

01 
02 
03 

314 
423 
531 

325 
433 
541 

336 
444 
552 

347 
455 
5()3 

a58 
4(J6 
574 

369 

477 
584 

379 

487 
595 

390 
498 
606 

401 
509 
617 

412 
520 
627 

04 
05 

06 

638 
746 
853 

649 
756 
863 

660 

767 
874 

670 

778 
885 

681 
788 
895 

692 

799 
906 

703 
810 
917 

713 
821 
927 

724 

831 
938 

735 
842 
94!) 

07 
08 
09 

959 

610()() 

172 

970 
077 
183 

981 
087 
194 

991 

098 
204 

*002 
109 
215 

*013 
119 
225 

*023 
130 
23() 

*034 
140 
247 

*045 
151 

257 

*055 
1()2 

2()S 

410 

278 

289 

;300 

310 

321 

331 

342 

352 

363 

374 

11 
12 
13 

384 
490 
595 

395 
500 
606 

405 
511 
616 

416 
521 
627 

426 
532 
637 

437 
542 
648 

448 
553 
658 

458 
563 
669 

4()9 
574 
679 

479 
584 
690 

14 
15 
16 

700 

805 
909 

711 

815 
920 

721 
826 
930 

731 
836 
941 

742 

847 
951 

752 

857 
962 

763 

868 
972 

773 

878 
982 

784 
888 
993 

794 

899 
*003 

17 

18 
19 

62014 
118 
221 

024 

128 
232 

034 
138 
242 

045 
149 
252 

055 
159 
263 

0f>6 
170 
273 

076 

180 

284 

086 
190 
294 

097 
201 
304 

107 
211 
315 

420 

325 

335 

346 

356 

366 

377 

387 

397 

408 

418 

21 
22 
23 

428 
531 
634 

439 
542 
644 

449 
552 
655 

459 
562 
()65 

469 
572 
675 

480 
583 
685 

490 
593 
696 

500 
603 
706 

511 
613 
716 

521 
(>24 
726 

1 
2 

11 

1.1 
2.2 

10 

1.0 
2.0 

9 

0.9 

1.8 

24 
25 
26 

737 
839 
941 

747 
849 
951 

757 
859 
961 

767 
870 
972 

778 
880 
982 

788 
890 
992 

798 

900 

*002 

808 

910 

*012 

818 

921 

*022 

829 

931 

*033 

3 
4 

5 
(\ 

3.3 
4.4 
5.5 

6.6 

7.7 
8.8 
9.9 

3.0 
4.0 
5.0 
6.0 
7.0 
8.0 
9.0 

2.7 
3.6 
4.5 
5.4 
6.3 
7.2 
8.1 

27 
28 
29 

63043 
144 
246 

053 
155 
256 

063 
165 

266 

073 
175 
276 

083 
185 
286 

094 
195 
2<H) 

104 
205 
306 

114 
215 
317 

124 

225 
327 

i;34 
236 
337 

7 
8 
9 

430 

347 

357 

3(57 

377 

387 

397 

407 

417 

428 

438 

31 
32 
33 

448 
548 
649 

458 
558 
659 

468 
568 
669 

478 
579 
679 

488 
589 
689 

498 
599 
699 

508 
609 
709 

518 
619 
719 

528 
629 
729 

538 
639 
739 

34 
35 

36 

749 
849 
949 

759 
859 
959 

769 
869 
969 

779 
879 
979 

789 
889 
988 

799 
899 
998 

809 

i)09 

*008 

819 

919 

*018 

829 

929 

*028 

839 
939 

*038 

37 
38 
39 

64048 
147 
246 

058 
157 
256 

068 
167 
266 

078 
177 
276 

088 

187 
286 

098 
197 
29() 

108 
207 
30() 

118 
217 
316 

128 
227 
32(i 

137 
237 
335 

440 

345 

355 

365 

375 

385 

395 

404 

414 

424 

434 

41 

42 
43 

444 
542 
640 

454 
552 
650 

464 

562 
660 

473 
572 
670 

483 
582 
680 

493 
591 
689 

503 
601 
699 

513 
611 
709 

523 
621 
719 

532 
(;31 
729 

44 

45 
46 

738 
836 
933 

748 
846 
943 

758 
856 
953 

768 

mo 

963 

777 
875 
972 

787 
885 
982 

797 
895 
992 

807 

iK)4 

*002 

816 

914 

*011 

826 

924 

*021 

47 

48 
49 

65031 
128 
225 

040 
137 
2:34 

050 
147 
244 

060 
157 
254 

070 
167 
263 

079 
176 
273 

089 
18() 
283 

099 

292 
389 

108 
205 
302 

118 
215 
312 

450 

321 

331 

341 

350 

3()0 

369 

379 

398 

408 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Prop.  Pts. 

\ 


T.able  3.    Number  Logarithiiis 


185 


0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Prop.  Pts. 

450 

51 
52 
53 

(i5  321 

331 

311 

350 

3(50 

369 

379 

389 

398 

408 

418 
514 
610 

427 
523 
619 

437 
533 
629 

447 
543 
639 

456 
552 
648 

466 
562 
658 

475 
571 
667 

485 
581 
677 

495 
591 
686 

504 
600 
696 

54 
55 
5G 

706 
801 
896 

715 

811 
90() 

725 
820 
91(3 

734 

8;^ 

925 

744 

839 
935 

753 
849 
944 

763 
858 
954 

772 
8(58 
963 

782 
877 
973 

792 
887 
982 

57 
58 
59 

992 

66  087 
181 

*001 
191 

*011 
106 
200 

*020 
115 
210 

*030 
124 
219 

*039 
134 
229 

*049 
143 

238 

*058 
153 
247 

*068 
162 
257 

*077 
172 
2(56 

460 

276 

285 

295 

304 

314 

323 

332 

342 

351 

361 

61 
62 
63 

370 
464 

558 

380 
474 
567 

389 
483 
577 

398 
492 
586 

408 
502 
59() 

417 
511 
605 

427 
521 
614 

436 

5;^ 

624 

445 
539 
633 

455 
549 
642 

64 
65 
6() 

652 
745 
839 

661 
755 
848 

671 

764 
857 

680 
773 
867 

689 
783 
876 

(599 
792 
885 

708 
801 
894 

717 
811 
904 

727 
820 
913 

736 
829 
922 

67 

68 

932 

67  025 
117 

941 
034 
127 

950 
043 
i;i6 

960 
052 
145 

9()9 
062 
154 

978 
071 
1(54 

987 
080 
173 

997 
089 
182 

*006 
0<)9 
191 

*015 
108 
201 

470 

210 

219 

228 

237 

247 

256 

265 

274 

284 

293 

71 

72 
73 

302 
3f)4 
486 

311 
403 
495 

321 
413 

504 

330 
422 
514 

339 
431 
523 

348 
440 
532 

357 
449 
541 

367 
459 
550 

376 
468 
560 

385 
477 
569 

1 
2 

10 

1.0 
2.0 

9 

0.9 
1.8 

8 

0.8 
1.6 

74 
75 
76 

578 
669 
761 

587 
679 
770 

596 
688 

779 

605 
697 
788 

614 

706 
797 

624 
715 
806 

633 
724 
815 

642 

733 
825 

651 

742 
834 

660 

752 
843 

3 
4 
5 
6 

3.0 
4.0 
5.0 
60 

2.7 
3.6 
4.5 
54 

2.4 
3.2 
4.0 

4.8 

77 
78 
79 

480 

852 

943 

68  034 

861 
"952 
043 

870 
9(51 
052 

879 
970 
0(>1 

888 
979 
070 

897 
988 
079 

906 
997 

088 

916 

*00() 

097 

925 

*015 

106 

934 

*024 

115 

7 
8 
9 

7.0 

8.0 
9.0 

6.3 

7.2 
8.1 

5.6 
6.4 

7.2 

124 

133 

142 

151 

1()0 

169 

178 

187 

196 

205 

81 
82 
83 

215 
305 
395 

224 
314 
404 

233 
323 
413 

242 
332 
422 

251 
341 
431 

260 
350 
440 

269 
359 
449 

278 
368 
458 

287 
377 
467 

296 
386 
476 

84 

85 
86 

485 
574 
664 

494 

583 
673 

502 

592 
681 

511 

601 
690 

520 
610 
699 

529 
619 

708 

538 

628 

717 

547 
(537 
726 

556 
646 

735 

565 
655 
744 

87 
88 
89 

753 
842 
931 

762 
851 
940 

771 

8(K) 
949 

780 
869 
958 

789 
878 
9(56 

797 
886 
975 

806 
895 
984 

815 
904 
9*)3 

824 

913 

*002 

833 

922 

*011 

490 

69  020 

028 

037 

046 

055 

064 

073 

082 

090 

099 

91 
92 
93 

108 
197 
285 

117 
205 
294 

126 
214 

302 

135 
223 
311 

144 
232 
320 

152 
241 
329 

161 
249 
338 

170 
258 
346 

179 
267 
355 

188 
276 
364 

94 
95 
96 

373 

548 

381 
469 
557 

390 
478 
5()6 

S99 
487 
574 

408 
496 
583 

417 
504 
592 

425 
513 
601 

434 
522 
609 

443 
531 
618 

452 
539 
627 

97 
98 
99 

636 
723 
810 

644 

732 
819 

653 
740 

827 

662 
749 
8;36 

671 

758 
845 

679 
767 
854 

688 

775 
862 

697 
784 
871 

705 
793 
880 

714 

801 

888 

500 

897 

906 

914 

923 

932 

940 

949 

958 

966 

975 

0 

1 

2 

3 

4  ! 

5 

6 

7  1 

8 

9 

Prop.  Pts. 

186 


Table  3.    Number  Logarithms 


0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Prop.  Pts. 

500 

69  897 

906 

914 

923 

932 

940 

949 

958 

966 

975 

01 
02 
03 

984 

70  070 

157 

992 
079 
1G5 

*001 
088 
174 

*010 
183 

*018 
105 
191 

*027 
114 
200 

*036 
122 
209 

*044 
131 
217 

*053 
140 
226 

*062 
148 
234 

04 
05 
06 

243 
329 
415 

252 

338 
424 

260 
346 
432 

269 
355 
441 

278 
364 
449 

286 
372 
458 

295 
381 
467 

303 
389 
475 

312 

398 
484 

321 
406 
492 

07 
08 
09 

501 
586 
672 

509 
595 
680 

518 
fi03 
689 

526 
612 
697 

535 
621 
706 

544 
629 
714 

552 
638 
723 

561 
646 
731 

569 
655 
740 

578 
663 
749 

510 

757 

766 

774 

783 

791 

800 

808 

817 

825 

834 

11 
12 
13 

842 

927 

71012 

851 
935 
020 

859 
944 
029 

868 
952 
037 

876 
961 
046 

885 
9(J9 
054 

893 
978 
063 

902 
986 
071 

910 
995 
079 

919 

*003 
088 

14 
15 

16 

096 

181 
265 

105 
189 
273 

113 

198 
282 

122 
20(3 
290 

130 
214 
299 

139 
223 

307 

147 
231 
315 

155 
240 
324 

164 

248 
332 

172 
257 
341 

17 
18 
19 

349 
433 
517 

357 
441 
525 

366 
450 
533 

374 

458 
542 

383 
466 
550 

391 
475 
559 

399 
483 
567 

408 
492 
575 

416 

500 
584 

425 
508 
592 

520 

600 

609 

617 

625 

634 

642 

650 

659 

667 

675 

21 
22 

23 

684 

767 
850 

692 
775 

858 

700 
784 
867 

709 
792 

875 

717 

800 
883 

725 
809 
892 

734 
817 
900 

742 
825 
908 

750 
834 
917 

759 
842 
925 

1 

2 

9 

0.9 
1.8 

8 

0.8 
1.6 

7 

0.7 
1.4 

24 

25 
26 

933 

72016 

099 

941 
024 
107 

950 
032 
115 

958 
041 
123 

966 
049 
132 

975 
057 
140 

983 
066 
148 

991 
074 
156 

999 
082 
165 

*008 
090 
173 

3 
4 
5 
6 

2.7 
3.6 
4.5 
5  4 

2.4 
3.2 
4.0 

4  8 

2.1 
2.8 
3.5 

4  2 

27 
28 
29 

181 
263 
346 

189 
272 
354 

198 

280 
362 

206 
288 
370 

214 

2iX> 

378 

222 
304 
387 

230 
313 
395 

239 
321 
403 

247 
329 
411 

255 
337 
419 

7 
8 
9 

6.3 
7.2 
8.1 

5.6 
6.4 
7.2 

4.9 
5.6 
6.3 

530 

428 

43() 

444 

452 

460 

469 

477 

485 

493 

501 

31 
32 
33 

509 
591 
673 

518 
599 
681 

526 
607 
689 

534 
616 

697 

542 
624 
705 

550 
632 
713 

558 
640 

722 

567 
648 
730 

575 
65(i 
738 

583 
665 
746 

34 
35 
36 

754 
835 
916 

762 
843 
925 

770 
852 
933 

779 
860 
941 

787 
868 
949 

795 
876 
957 

803 
884 
965 

811 
892 
973 

819 
900 
981 

827 
i)08 
989 

37 

38 
39 

997 

73078 

159 

*006 
086 
167 

*014 
094 
175 

*022 
102 
183 

*030 
111 
191 

*038 
119 
199 

*046 
127 
207 

*054 
135 
215 

*062 
143 
223 

*070 
151 
231 

540 

239 

247 

255 

263 

272 

•280 

288 

296 

304 

312 

41 
42 
43 

320 

400 
480 

328 
408 

488 

336 
416 
496 

344 
424 
504 

352 
432 
512 

.  360 
440 
520 

368 
448 
528 

376 
456 
536 

384 
464 
544 

392 
472 
552 

44 
45 
46 

560 
640 
719 

568 
648 

727 

576 
656 
735 

584 
6f)4 
743 

592 
672 
751 

600 
679 
759 

608 
687 
767 

616 
695 
775 

624 
703 
783 

632 
711 
791 

47 
48 
49 

799 
878 
957 

807 
886 
965 

815 
894 
973 

823 
902 
981 

830 
910 
989 

838 
918 
997 

846 

926 

*005 

854 

933 

*013 

092 

862 

941 

*020 

870 

949 

*028 

550 

74  0.% 

044 

052 

060 

068 

076 

084 

099 

107 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Prop.  PtB. 

Table 

3.  Number 

Log 

arithms 

187 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Prop.  Pts.    1 

560 

74  036 

044 

052 

0()0 

068 

076 

084 

092 

0*)9 

107 

51 
52 
53 

115 
194 

273 

123 
202 

280 

131 
210 

288 

139 

218 
296 

147 
225 
304 

155 
233 
312 

162 
241 
320 

170 
249 

327 

178 
257 
335 

186 
265 
343 

54 
55 
56 

351 
429 
507 

359 
437 
515 

367 
445 
523 

374 
453 
531 

382 
461 
539 

390 
468 
547 

398 
47() 
554 

406 
484 
562 

414 
492 
570 

421 

500 
578 

57 
58 
59 

586 
663 
741 

593 
671 
749 

601 
679 

757 

609 

687 
764 

617 

695 

772 

624 

702 
780 

632 

710 

788 

640 

718 
796 

648 
726 
803 

656 
733 
811 

560 

819 

827 

834 

842 

850 

858 

865 

873 

881 

889 

61 
62 
63 

896 

974 

75051 

904 
981 
059 

912 

989 
066 

920 
997 
074 

927 
*005 

082 

935 

*012 
089 

943 

*020 

097 

950 

*028 

105 

958 

*035 

113 

966 

*043 

120 

64 
65 
66 

128 
205 

282 

136 
213 

289 

143 
220 
297 

151 

228 
305 

159 
236 
312 

166 
243 
320 

174 
251 

328 

182 
259 
335 

189 
266 
343 

197 
274 
351 

67 
68 
69 

358 
435 
511 

366 
442 
519 

374 
450 
526 

381 

458 
534 

389 
465 
542 

397 
473 
549 

404 

481 
557 

412 

488 
565 

420 
496 

572 

427 
504 

580 

570 

587 

595 

603 

610 

618 

626 

633 

641 

648 

656 

71 
72 
73 

664 
740 
815 

671 

747 
823 

679 
755 
831 

686 
762 
838 

694 
770 
846 

702 
778 
853 

709 
785 
861 

717 
793 
868 

724 

800 
876 

732 

808 
884 

1 

2 

8 

0.8 
1.6 

7 

0.7 
1.4 

74 
75 
76 

891 
76042 

899 
974 
050 

906 
982 
057 

914 
989 
065 

921 

997 
072 

929 

*005 
080 

937 

*012 

087 

944 

*020 

095 

952 

*027 

103 

959 

*035 

110 

3 
4 
5 
6 

2.4 
3.2 
4.0 

4.8 

2.1 
2.8 
3.5 
4.2 

77 
78 
79 

118 
193 
268 

125 

200 
275 

133 

208 
283 

140 
215 

25K) 

148 
223 

298 

155 

230 
305 

163 
238 
313 

170 
245 
320 

178 
253 
328 

185 
260 
335 

7 
8 
9 

5.6 
6.4 

7.2 

4.9 
5.6 
6.3 

580 

343 

350 

358 

365 

373 

380 

388 

395 

403 

410 

81 
82 
83 

418 
492 
567 

425 
500 
574 

433 
507 

582 

440 
515 

589 

448 
522 
597 

455 
530 
604 

462 
537 
612 

470 
545 
619 

477' 

552 

626 

485 
559 
634 

84 
85 
86 

641 

716 
790 

649 
723 

797 

656 

730 
805 

664 
738 
812 

671 
745 

819 

678 
753 

827 

686 
760 
834 

693 

768 
842 

701 

775 
849 

708 
782 
856 

87 
88 
89 

864 

938 

77  012 

871 
945 
019 

879 
953 
026 

886 
960 
034 

893 
i)67 
041 

901 
975 
048 

908 
982 
056 

916 

989 
063 

923 
997 
070 

930 

*004 

078 

590 

085 

093 

100 

107 

115 

122 

129 

137 

144 

151 

91 
92 
93 

159 
232 
305 

166 
240 
313 

173 
247 
320 

181 
254 
327 

188 
262 
335 

195 
269 
342 

203 
276 
349 

210 

283 
357 

217 
291 
364 

225 
298 
371 

94 
95 
96 

379 
452 
525 

386 
459 
532 

393 
466 
539 

401 
474 
546 

408 
481 
554 

415 

488 
561 

422 

495 
568 

430 
503 
576 

437 
510 
583 

444 
517 
590 

97 
98 
99 

597 
670 
743 

605 
677 
750 

612 

685 
757 

619 
692 
764 

627 
699 

772 

634 

706 
779 

641 
714 

786 

648 
721 
793 

656 

728 
801 

663 
735 

808 

600 

815 

822 

830 

837 

844 

851 

859 

866 

873 

880 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Prop.  Pts. 

188 


Table  3.    Number  Logarithms 


0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Prop,  Pts. 

600 

77  815 

822 

830 

837 

844 

851 

859 

8(56 

873 

880 

01 

887 

895 

902 

909 

916 

924 

931 

938 

945 

952 

02 

960 

mi 

974 

981 

988 

996 

*003 

*010 

*017 

*025 

03 

78032 

039 

046 

053 

061 

068 

075 

082 

089 

097 

04 

104 

111 

118 

125 

132 

140 

147 

154 

161 

168 

05 

176 

183 

190 

197 

204 

211 

219 

226 

233 

240 

OG 

247 

254 

262 

269 

276 

283 

290 

297 

305 

312 

07 

319 

326 

333 

340 

347 

355 

m2 

369 

376 

383 

08 

390 

398 

405 

412 

419 

426 

433 

440 

447 

455 

09 

462 

4(J9 

476 

483 

490 

497 

504 

512 

519 

526 

610 

533 

540 

547 

554 

561 

569 

576 

583 

590 

597 

11 

604 

611 

618 

625 

633 

640 

647 

654 

(561 

668 

12 

675 

682 

689 

696 

704 

711 

718 

725 

732 

739 

13 

746 

753 

760 

767 

774 

781 

789 

796 

803 

810 

14 

817 

824 

831 

838 

845 

852 

859 

866 

873 

880 

15 

888 

85)5 

902 

909 

916 

923 

930 

937 

du 

951 

16 

958 

965 

972 

979 

986 

993 

^'000 

*007 

*014 

*021 

17 

79029 

036 

043 

050 

057 

064 

071 

078 

085 

092 

18 

099 

106 

113 

120 

127 

134 

141 

148 

155 

162 

19 

169 

176 

183 

190 

197 

204 

211 

218 

225 

232 

620 

239 

246 

253 

260 

267 

274 

281 

288 

295 

302 

21 

309 

316 

323 

330 

337 

344 

351 

358 

365 

372 

8 

7 

6 

22 

379 

386 

393 

400 

407 

414 

421 

428 

435 

442 

1 

0.8 

0.7 

0.6 

23 

449 

456 

463 

470 

477 

484 

491 

498 

505 

511 

2 

1.6 

1.4 

1.2 

24 

518 

525 

532 

539 

546 

553 

560 

567 

574 

581 

3 
4 
5 
6 

2.4 
3.2 
4.0 

4.8 

2.1 
2.8 
3.5 
4.2 

1.8 
2.4 
3.0 
3.6 

25 

588 

595 

602 

609 

616 

623 

630 

637 

644 

650 

26 

657 

664 

671 

678 

685 

692 

699 

706 

713 

720 

27 

727 

734 

741 

748 

754 

761 

768 

775 

782 

789 

7 

5.6 

4.9 

4.2 

28 

796 

803 

810 

817 

824 

831 

837 

844 

851 

858 

8 

6.4 

5.6 

4.8 

29 

865 

872 

879 

886 

893 

900 

iKX) 

913 

920 

927 

y 

7.2 

6.3 

5.4 

630 

934 

941 

948 

955 

962 

im 

975 

982 

989 

95)6 

31 

80003 

010 

017 

024 

030 

037 

044 

051 

058 

065 

32 

072 

079 

085 

092 

099 

106 

113 

120 

127 

134 

33 

140 

147 

154 

161 

168 

175 

182 

188 

195 

202 

34 

209 

216 

223 

229 

236 

243 

250 

257 

264 

271 

35 

277 

284 

291 

298 

305 

312 

318 

325 

332 

339 

36 

346 

353 

359 

366 

373 

380 

387 

393 

400 

407 

37 

414 

421 

428 

434 

441 

448 

455 

4(52 

468 

475 

38 

482 

489 

496 

502 

509 

516 

523 

530 

536 

543 

39 

550 

557 

564 

570 

577 

584 

591 

598 

604 

611 

640 

618 

625 

632 

638 

645 

652 

659 

665 

(572 

(579 

41 

686 

693 

699 

706 

713 

720 

726 

733 

740 

747 

42 

754 

760 

767 

774 

781 

787 

794 

801 

808 

814 

43 

821 

828 

835 

841 

848 

855 

862 

868 

875 

882 

44 

889 

895 

902 

909 

916 

922 

929 

936 

943 

945) 

45 

956 

9(53 

t)69 

976 

983 

990 

99(i 

*003 

*010 

*017 

46 

81023 

030 

037 

043 

050 

057 

064 

070 

077 

084 

47 

090 

097 

104 

111 

117 

124 

131 

137 

144 

151 

48 

158 

164 

171 

178 

184 

191 

198 

204 

211 

218 

49 

224 

231 

238 

245 

251 

258 

265 

271 

278 

285 

650 

291 

298 

305 

311 

318 

325 

331 

338 

a45 

3;51 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Prop.  Pts. 

i 


Table  3.  Number  Log 

arithms 

189 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Prop.  Pts. 

650 

81  291 

298 

305 

311 

318 

325 

331 

338 

345 

351 

51 

358 

3(35 

371 

378 

385 

391 

398 

405 

411 

418 

52 

425 

431 

438 

445 

451 

458 

465 

471 

478 

485 

53 

491 

498 

505 

511 

518 

525 

531 

538 

544 

551 

54 

558 

564 

571 

578 

584 

591 

598 

604 

611 

617 

55 

624 

631 

637 

644 

()51 

657 

()(;4 

671 

677 

(i84 

56 

690 

697 

704 

710 

717 

723 

730 

737 

743 

750 

57 

757 

763 

770 

776 

783 

790 

79(3 

803 

809 

816 

58 

823 

829 

sm 

842 

849 

856 

862 

869 

875 

882 

59 

889 

895 

902 

908 

915 

921 

928 

935 

941 

948 

660 

954 

961 

9f)8 

974 

981 

987 

994 

*000 

*()07 

*014 

61 

82  020 

027 

033 

040 

046 

053 

060 

066 

073 

079 

62 

086 

092 

0i)9 

105 

112 

119 

125 

132 

138 

145 

63 

151 

158 

164 

171 

178 

184 

191 

197 

204 

210 

64 

217 

223 

230 

236 

243 

249 

256 

263 

269 

276 

65 

282 

289 

295 

302 

308 

315 

321 

328 

334 

341 

m 

347 

354 

360 

367 

373 

380 

387 

393 

400 

406 

67 

413 

419 

426 

432 

439 

445 

452 

458 

465 

471 

68 

478 

484 

491 

497 

504 

510 

517 

523 

530 

536 

69 

543 

549 

556 

562 

569 

575 

582 

5S8 

595 

601 

670 

607 

614 

620 

627 

(333 

640 

646 

653 

659 

6()() 

71 

672 

679 

685 

692 

698 

705 

711 

718 

724 

730 

7 

6 

72 

737 

743 

750 

756 

763 

769 

77(; 

782 

789 

795 

1 

0.7 

0.6 

73 

802 

808 

814 

821 

827 

834 

840 

847 

853 

8(30 

2 

1.4 

1.2 

74 

866 

872 

879 

885 

892 

898 

905 

911 

918 

924 

3 
4 
5 
6 

2.1 

2.8 
3.5 
4.2 

1.8 
2.4 
30 
3.6 

75 

930 

937 

943 

950 

956 

963 

969 

975 

982 

988 

76 

995 

*001 

*008 

*014 

*020 

*027 

*033 

*040 

*046 

*052 

77 

83  059 

065 

072 

078 

085 

091 

097 

104 

110 

117 

7 

4.9 

4.2 

78 

123 

129 

136 

142 

149 

155 

161 

168 

174 

181 

8 

5.6 

4.8 

79 

187 

193 

200 

206 

213 

219 

225 

232 

238 

245 

9 

6.3 

5.4 

680 

251 

257 

264 

270 

276 

283 

289 

296 

302 

308 

81 

315 

321 

327 

334 

340 

347 

353 

359 

366 

372 

82 

378 

385 

391 

398 

404 

410 

417 

423 

429 

436 

83 

442 

448 

455 

461 

467 

474 

480 

487 

493 

499 

84 

506 

512 

518 

525 

531 

537 

544 

550 

556 

563 

85 

569 

575 

582 

588 

5i)4 

601 

607 

(313 

()20 

62(3 

86 

632 

639 

645 

651 

658 

664 

670 

677 

683 

689 

87 

696 

702 

708 

715 

721 

727 

734 

740 

746 

753 

88 

759 

7(55 

771 

778 

784 

790 

797 

803 

809 

816 

89 

822 

828 

835 

841 

847 

853 

8(30 

8(36 

872 

879 

690 

885 

891 

897 

904 

910 

916 

923 

929 

935 

942 

91 

948 

954 

960 

967 

973 

979 

985 

992 

998 

*004 

92 

84011 

017 

023 

029 

036 

042 

048 

055 

061 

067 

93 

073 

080 

086 

092 

098 

105 

111 

117 

123 

130 

94 

136 

142 

148 

155 

161 

167 

173 

180 

186 

192 

95 

198 

205 

211 

217 

223 

230 

236 

242 

248 

255 

96 

261 

267 

273 

280 

286 

292 

298 

305 

311 

317 

97 

323 

330 

336 

342 

348 

354 

361 

367 

373 

379 

98 

386 

392 

398 

404 

410 

417 

423 

429 

435 

442 

i)9 

448 

454 

4()0 

466 

473 

479 

485 

491 

497 

504 

700 

510 

516 

522 

528 

535 

541 

547 

553 

559 

566 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Prop.  Pts. 

190 

Table  3. 

Vumber  Logarithms 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Prop.  Pts. 

700 

84  510 

516 

522 

528 

535 

541 

547 

553 

559 

566 

01 

572 

578 

584 

590 

597 

603 

609 

615 

621 

628 

02 

634 

(yiO 

646 

652 

658 

665 

671 

677 

683 

689 

03 

69(3 

702 

708 

714 

720 

726 

733 

739 

745 

751 

04 

757 

7()3 

770 

776 

782 

788 

794 

800 

807 

813 

05 

819 

825 

831 

837 

844 

850 

856 

862 

868 

874 

06 

880 

887 

893 

899 

905 

911 

917 

924 

930 

936 

07 

942 

948 

954 

9()0 

967 

973 

979 

985 

991 

997 

08 

85003 

009 

016 

022 

028 

034 

040 

046 

052 

058 

09 

065 

071 

077 

083 

089 

095 

101 

107 

114 

120 

710 

126 

132 

138 

144 

150 

156 

163 

169 

175 

181 

11 

187 

193 

199 

205 

211 

217 

224 

230 

236 

242 

12 

248 

254 

260 

266 

272 

278 

285 

291 

297 

303 

13 

309 

315 

321 

327 

333 

339 

345 

352 

358 

364 

14 

370 

376 

382 

388 

394 

400 

406 

412 

418 

425 

15 

431 

437 

443 

449 

455 

461 

467 

473 

479 

485 

16 

491 

497 

503 

509 

516 

522 

528 

534 

540 

546 

17 

552 

558 

564 

570 

576 

582 

588 

594 

600 

606 

18 

612 

618 

625 

631 

637 

643 

649 

655 

661 

667 

19 

673 

679 

685 

691 

697 

703 

709 

715 

721 

727 

720 

733 

739 

745 

751 

757 

763 

769 

775 

781 

788 

21 

794 

800 

806 

812 

818 

824 

830 

836 

842 

848 

7 

6 

5 

22 

854 

8(>0 

866 

872 

878 

884 

890 

896 

902 

<X)8 

1 

0.7 

0.6 

0.5 

23 

914 

920 

926 

932 

938 

944 

950 

956 

962 

968 

2 

1.4 

1.2 

1.0 

24 

974 

980 

986 

992 

998 

*004 

*010 

*016 

*022 

*028 

3 
4 
5 
6 

2.1 

2.8 
3.5 
4.2 

1.8 
2.4 
3.0 
3.6 

1.5 
2.0 
2.5 
3.0 

25 

86034 

040 

046 

052 

058 

064 

070 

076 

082 

088 

26 

094 

100 

106 

112 

118 

124 

130 

136 

141 

147 

27 

153 

159 

165 

171 

177 

183 

189 

195 

201 

207 

7 

4.9 

4.2 

3.5 

28 

213 

219 

225 

231 

237 

243 

249 

255 

261 

2()7 

8 

5.6 

4.8 

4.0 

29 

273 

279 

285 

291 

297 

303 

308 

314 

320 

326 

9 

6.3 

5.4 

4.5 

730 

332 

338 

344 

350 

356 

362 

3(38 

374 

380 

386 

31 

392 

398 

404 

410 

415 

421 

427 

433 

439 

445 

32 

451 

457 

463 

469 

475 

481 

487 

493 

499 

504 

33 

510 

516 

522 

528 

534 

540 

546 

552 

558 

564 

34 

570 

576 

581 

587 

593 

599 

605 

611 

617 

623 

35 

629 

635 

641 

646 

652 

658 

6(>4 

670 

676 

682 

36 

688 

694 

700 

705 

711 

717 

723 

729 

735 

741 

37 

747 

753 

759 

764 

770 

776 

782 

788 

794 

800 

38 

80<5 

812 

817 

823 

829 

835 

841 

847 

853 

859 

39 

864 

870 

876 

882 

888 

894 

SXX) 

^m 

911 

917 

740 

923 

929 

935 

941 

947 

953 

958 

964 

970 

976 

41 

982 

988 

994 

999 

*005 

*011 

*017 

*023 

*029 

*035 

42 

87  040 

046 

052 

058 

064 

070 

075 

081 

087 

093 

43 

099 

105 

111 

116 

122 

128 

134 

140 

146 

151 

44 

157 

163 

169 

175 

181 

186 

192 

198 

204 

210 

45 

216 

221 

227 

233 

239 

245 

251 

25(3 

262 

268 

46 

274 

280 

286 

291 

297 

303 

309 

315 

320 

326 

47 

332 

338 

344 

349 

355 

361 

367 

373 

379 

384 

48 

390 

3<)6 

402 

408 

413 

419 

425 

431 

437 

442 

49 

448 

454 

460 

466 

471 

477 

483 

489 

495 

500 

750 

506 

512 

518 

523 

529 

535 

541 

547 

552 

558 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Prop.  Pts. 

Table  3.    Number  Logarithms 


191 


0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Prop.  Pts. 

750 

51 
52 
53 

54 
55 
56 

57 
58 
59 

87  506 

512 

518 

52-A 

529 

535 

541 

547 

552 

558 

564 
622 
679 

737 
795 

852 

910 

mi 

88  024 

570 

628 
685 

743 

800 
858 

915 
973 
030 

576 
633 
691 

749 

8O0P 
864 

921 
978 
03() 

581 
639 
697 

754 
812 
869 

927 
984 
041 

587 
645 
703 

760 
818 
875 

933 
990 
047 

593 
651 
708 

766 

823 
881 

938 
996 
053 

599 
65() 
714 

772 
829 
887 

944 

*001 

058 

604 
662 
720 

777 
835 
892 

950 

*007 

064 

610 
6()8 
726 

783 
841 
898 

955 

*013 

070 

616 
674 
731 

789 
846 
904 

961 

*018 
076 

760 

081 

087 

093 

098 

104 

110 

116 

121 

127 

133 

61 
62 
63 

64 
65 

66 

67 

68 
69 

138 
195 
252 

309 
3(56 
423 

480 
536 
593 

144 
201 

258 

315 
372 
429 

485 
542 
598 

150 
207 
264 

321 
377 
434 

491 
547 
604 

156 
213 
270 

326 
383 
440 

497 
553 
610 

161 
218 
275 

332 
389 
446 

502 
559 
615 

167 
224 

281 

338 
395 
451 

508 
564 
621 

173 
230 

287 

343 
400 
457 

513 

570 
627 

178 
235 
292 

349 
406 
463 

519 

576 
632 

184 
241 

298 

355 
412 
468 

525 
581 
(538 

190 
247 
304 

360 
417 
474 

530 
587 
643 

770 

649 

655 

660 

666 

672 

677 

683 

689 

694 

700 

71 

72 
73 

7-t 
75 
76 

77 
78 
79 

705 
762 

818 

874 
930 
986 

89042 
098 
1.54 

711 

767 
824 

880 
936 
992 

048 
104 
159 

717 
773 
829 

885 
941 
997 

053 
109 
1()5 

722 
779 
835 

891 

947 

*003 

059 
115 
170 

728 
784 
840 

897 

953 

*009 

064 
120 
176 

734 
790 
846 

902 

958 

*014 

070 
126 
182 

739 
795 
852 

908 

9(54 

*020 

076 
131 

187 

745 
801 
857 

913 

md 

*025 

081 
137 
193 

750 
807 
863 

919 

975 
•*031 

087 
143 
198 

756 
812 
868 

925 

981 

*037 

092 
148 
204 

1 

2 
3 
4 
5 
6 
7 
8 
9 

6 

0.6 
1.2 
1.8 
2.4 
3.0 
3.6 
4.2 
4.8 
5.4 

5 

0.5 
1.0 
1.5 
2.0 
2.5 
3.0 
3.5 
4.0 
4.5 

780 

20<) 

215 

221 

226 

232 

237 

243 

248 

254 

260 

81 
82 
83 

84 

85 
86 

87 
88 
89 

265 
321 
376 

432 
487 
542 

597 
653 

708 

271 
326 

382 

437 
492 
548 

603 
658 
713 

276 
332 
387 

443 
498 
553 

609 
664 
719 

282 
337 
393 

448 
504 
559 

614 
669 
724 

287 
343 
398 

454 
509 
564 

620 

675 
730 

293 
348 
404 

459 
515 
570 

625 
680 
735 

298 
354 
409 

465 
520 
575 

631 

686 
741 

304 
360 
415 

470 
526 
581 

636 

691 

'746 

310 
365 
421 

476 
531 
586 

642 

697 
752 

315 
371 
426 

481 

537 
592 

647 
702 

757 

790 

763 

768 

774 

779 

785 

790 

796 

801 

807 

812 

91 
92 
93 

94 
95 
96 

97 

98 
99 

818 
873 
927 

982 

90037 

091 

146 

200 
255 

823 
878 
933 

988 
042 
097 

151 
206 
260 

829 
883 
938 

993 
048 
102 

157 
211 
266 

834 
889 
944 

998 
053 
108 

162 
217 
271 

840 
894 
949 

*004 
059 
113 

168 
222 
276 

845 
900 
955 

*009 
064 
119 

173 

227 
282 

851 
W5 
960 

*015 
069 
124 

179 
233 

287 

856 
911 
966 

*020 
075 
129 

184 
238 
293 

862 
916 
971 

*026 
080 
135 

189 
244 

298 

867 
922 
977 

*031 
086 
140 

195 
249 
304 

800 

309 

314 

320 

325 

331 

336 

342 

347 

352 

358 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Prop.  Pts. 

192 


Table  3.    Number  Loffarithms 


0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Prop.  Pts. 

800 

90  .'309 

314 

320 

325 

331 

336 

342 

M7 

352 

358 

01 
02 
03 

3(i3 
417 
472 

3(59 
423 

477 

374 

428 
482 

380 
434 
488 

385 
439 
493 

390 
445 
499 

3f)6 
450 
504 

401 
455 
509 

407 
461 
515 

412 
466 
520 

04 
05 
06 

526 

580 
634 

531 

585 
639 

536 
590 
644 

542 
650 

547 
(iOl 
655 

553 

607 
660 

558 
612 

me 

563 
617 
671 

5(59 
623 
677 

574 
638 

682 

07 

08 
09 

687 
741 
795 

693 
747 
800 

698 
752 
806 

703 

757 
811 

709 
763 
816 

714 

768 
822 

720 
773 
827 

725 
779 
832 

7.30 

784 
838 

7,36 
789 
843 

810 

849 

8.54 

859 

865 

870 

875 

881 

886 

891 

897 

11 
12 
13 

902 

95() 

91009 

907 
961 
014 

913 
966 
020 

918 
972 
025 

924 

977 
030 

929 

982 

.036 

934 
988 
041 

940 
993 
046 

945 
998 
052 

950 

*004 

057 

14 
15 
16 

062 
116 
169 

068 
121 
174 

073 
12(5 
180 

078 
132 
185 

084 
137 
190 

089 
142 
196 

OM 
148 
201 

100 
153 
206 

105 

1.58 
212 

110 
1(54 
217 

17 
18 
19 

222 
275 

328 

228 
281 
3,34 

233 

286 
339 

238 
291 
344 

243 

297 
350 

249 
302 
a55 

254 
307 
3(50 

259 
312 
365 

265 
318 
371 

270 
323 

376 

820 

381 

387 

392 

397 

403 

408 

413 

418 

424 

429 

21 
22 

23 

434 

487 
540 

440 
492 
545 

445 
498 
551 

450 
503 
556 

455 
508 
561 

461 
514 
566 

4(5(5 
519 
572 

471 
524 

577 

477 
529 
582 

482 
535 
587 

1 
2 

6 

0.6 
1.2 

5 

0.5 
1.0 

24 
25 

26 

593 
645 

698 

598 
651 
703 

cm 

709 

609 
(5(51 
714 

614 
(5(5(5 
719 

619 
(572 
724 

624 

(577 
730 

630 

682 
735 

635 
687 
740 

640 
693 
745 

3 
4 
5 
6 

1.8 
2.4 
3.0 
3.6 

1.5 
2.0 
2.5 
3.0 

27 

28 
29 

751 
803 

855 

756 
808 
8(51 

761 
814 

866 

766 

819 

871 

772 
824 
876 

777 
829 
882 

782 
8.34 

887 

787 
840 
8i)2 

793 
845 
897 

798 
850 
903 

7 
8 
9 

4.2 
4.8 
5.4 

3.5 
4.0 
4.5 

830 

fK)8 

913 

918 

924 

929 

934 

9.39 

944 

950 

955 

31 

32 
33 

f)60 

92  012 

065 

965 
018 
070 

971 
023 
075 

976 

028 
080 

981 
033 
085 

986 
038 
091 

991 
044 
096 

997 
049 
101 

*002 
054 
106 

*007 
059 
111 

34 
35 
36 

117 
1(J9 
221 

122 
174 
226 

127 
179 
231 

132 

184 
236 

137 
189 
241 

143 
195 

247 

148 
200 
252 

153 
205 
257 

158 
210 
262 

163 
215 
267 

37 
38 
39 

273 
324 
376 

278 
330 
381 

283 
335 

387 

288 
340 
392 

293 

345 
397 

298 
350 
402 

304 

a55 

407 

309 
3(51 
412 

314 

3(5(5 
418 

319 
371 
423 

840 

428 

433 

438 

443 

449 

454 

459 

464 

4(59 

474 

41 
42 

43 

480 
531 
583 

485 

588 

490 
542 
593 

495 

547 
598 

500 
552 
603 

.505 
557 
609 

511 

562 
614 

516 
5(57 
619 

521 
.572 
624 

526 
578 
629 

44 

45 
46 

634 

686 
737 

639 
(591 
742 

645 
696 

747 

650 
701 
752 

655 
706 
758 

660 
711 
763 

665 
716 
768 

670 

722 
773 

675 

727 
778 

(581 
732 
783 

47 

48 
49 

788 
840 
891 

793 
845 

896 

799 

mo 

901 

804 
855 
906 

809 
8(50 
911 

814 
865 
916 

819 
870 
921 

824 

875 
927 

829 
881 
932 

8,34 
886 
937 

850 

942 

947 

952 

957 

9(52 

967 

973 

978 

983 

988 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Prop.  Pts. 

Table  3.    Number  Logarithms 


193 


0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Prop.  Pts. 

850 

92  942 

947 

952 

957 

962 

967 

973 

978 

983 

988 

51 
52 
53 

993 

93044 

095 

998 
049 
100 

*003 
054 
105 

*008 
059 
110 

*013 
064 
115 

*018 
0(59 
120 

*024 
075 
125 

*029 
080 
131 

*034 
085 
136 

*039 
O^K) 
141 

54 
55 
56 

146 
197 
247 

151 
202 
252 

156 

207 
258 

161 
212 
263 

166 
217 
268 

171 
222 
273 

176 
227 

278 

181 
232 

283 

186 
237 

288 

192 
242 

293 

57 
58 
59 

298 
349 
399 

303 
354 
404 

308 
359 
409 

313 
364 
414 

318 
369 
420 

323 
374 
425 

328 
379 
430 

334 
384 
435 

339 
389 
440 

M4 
394 
445 

860 

450 

455 

460 

465 

470 

475 

480 

485 

490 

495 

61 
62 
63 

500 
551 
601 

505 

556 
606 

510 
561 
611 

515 

56G 
616 

520 
571 
621 

526 
576 
626 

531 

581 
631 

536 
586 
636 

541 
591 
641 

546 
596 
646 

64 
65 

(y6 

651 

702 
752 

656 
707 
757 

661 
712 

762 

666 
717 

767 

671 

722 
772 

676 

727 
777 

682 
732 

782 

687 
737 

787 

692 
742 
792 

697 
747 

797 

67 
68 
69 

802 
852 
902 

807 
857 
907 

812 
862 
912 

817 
867 
917 

822 

872 
922 

827 
877 
927 

832 
882 
932 

837 
887 
937 

842 
892 
942 

847 
897 
947 

870 

952 

957 

962 

f)67 

972 

977 

982 

987 

992 

997 

71 

72 
73 

94  002 
052 
101 

007 
057 
106 

012 
0()2 
111 

017 
067 
116 

022 
072 
121 

027 
077 
126 

032 
082 
131 

037 
086 
136 

Oi2 
091 
141 

047 
096 
146 

1 
2 

6 

0.6 
1.2 

5 

0.5 
1.0 

4 

0.4 
0.8 

74 

75 
76 

151 
201 
250 

156 
206 
255 

161 
211 

260 

166 
216 
265 

171 
221 
270 

176 
226 
275 

181 
231 

280 

186 
236 
285 

191 

240 
290 

196 
245 
295 

3 
4 
5 

1.8 
2.4 
3.0 
3  6 

1.5 

2.0 
2.5 
30 

1.2 
1.6 
2.0 
2.4 

77 
78 
79 

300 
349 
399 

305 
354 
404 

310 
359 
409 

315 
364 
414 

320 
3(J9 
419 

325 

374 
424 

330 
379 
429 

335 
384 
433 

340 
389 
438 

345 
394 
443 

7 
8 
9 

4.2 
4.8 
5.4 

3.5 
4.0 
4.5 

2.8 
3.2 
3.6 

880 

448 

453 

458 

463 

468 

473 

478 

483 

488 

493 

81 
82 
83 

498 
547 
596 

503 
552 
601 

507 
557 
606 

512 
5()2 
611 

517 
5(>7 
616 

522 
571 
621 

527 
576 
626 

532 
581 
630 

537 
58() 
635 

542 
591 
640 

84 
85 
86 

645 
694 
743 

650 
699 
748 

655 
704 
753 

6(i0 
709 
758 

665 
714 

763 

670 
719 

768 

675 
724 

773 

680 
729 
778 

685 
7:54 
783 

689 
738 

787 

87 
88 
89 

792 
841 
890 

797 
84fj 
895 

802 
851 
900 

807 
856 
905 

812 
861 
910 

817 
8f>6 
915 

822 
871 
919 

827 
876 
924 

832 
880 
929 

836 
885 
934 

890 

939 

944 

949 

954 

959 

f)63 

968 

973 

978 

983 

91 
92 
93 

988 

95  om 

085 

993 
041 
090 

99S 
046 
095 

*002 
051 
100 

*007 
056 
105 

*012 
061 
109 

*017 
066 
114 

*022 
071 
119 

*t)27 
075 
124 

*032 
080 
129 

94 
95 
96 

134 
182 
231 

139 

187 
236 

143 

192 
240 

148 
197 
245 

153 
202 
250 

158 
207 
255 

163 
211 
260 

168 
216 
265 

173 
221 

270 

177 
226 
274 

97 
98 
99 

279 
328 
376 

284 
332 
381 

289 
337 
386 

294 
342 
390 

299 
347 
395 

303 
352 
400 

308 
357 
405 

313 
361 
410 

318 
366 
415 

323 
371 
419 

900 

424 

429 

434 

439 

444 

448 

453 

458 

463 

468 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Prop,  Pts. 

194 

Table  3.  Number 

Log 

arithnis 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Prop.  PtS: 

900 

95  424 

429 

434 

439 

444 

448 

453 

458 

463 

468 

01 
02 
03 

472 
521 
569 

477 
525 
574 

482 
530 
578 

487 
535 
583 

492 
540 

588 

497 
545 
593 

501 
550 
598 

506 
554 
602 

511 

559 
607 

516 
5(i4 
612 

04 

05 
06 

617 

665 
713 

622 

670 
718 

626 
674 

722 

631 
679 

727 

636 

684 
732 

641 

689 
737 

64(5 
694 
742 

650 
698 
746 

655 
703 
751 

(J(iO 
708 
756 

07 
08 
09 

761 
809 
85() 

766 
813 
861 

770 

818 
866 

775 
823 
871 

780 
828 
875 

785 
832 
880 

789 
837 
885 

794 
842 
890 

799 
847 
895 

804 
852 
899 

910 

904 

909 

914 

918 

923 

928 

933 

938 

942 

947 

11 
12 
13 

952 

999 

96.047 

957 

*004 

052 

961 

*009 

057 

966 

*014 

061 

971 

*019 

066 

976 

*023 

071 

980 

*028 

076 

985 

*033 

080 

990 

*038 

085 

995 

*042 

090 

14 
15 
16 

095 
142 
190 

099 
147 
194 

104 
152 
199 

109 
156 
204 

114 
161 
209 

118 
166 
213 

123 
171 

218 

128 
175 
223 

133 
180 
227 

137 
185 
232 

17 
18 
19 

237 

284 
332 

242 

289 
336 

246 
294 
341 

251 

298 
346 

256 
303 
350 

261 
308 
355 

265 
313 
360 

270 
317 
365 

275 
322 
369 

280 
327 
374 

920 

379 

384 

388 

393 

398 

402 

407 

412 

417 

421 

21 
22 
23 

426 
473 
520 

431 
478 
525 

4a5 

483 
530 

440 
487 
534 

445 
492 
539 

450 
497 
544 

454 
501 
548 

459 
506 
553 

464 
511 
558 

468 
515 
562 

1 

2 

5 

0.5 
1.0 

4 

0.4 

0.8 

24 

2,5 
26 

567 
614 

mi 

572 
619 
666 

577 
624 
670 

581 
628 
675 

586 
633 
680 

691 

(538 
685 

595 
642 
689 

600 
647 
694 

605 
652 
699 

609 
656 
703 

3 
4 

5 
6 

1.5 
2.0 
2.5 
3.0 

1.2 
1.6 
2.0 
2.4 

27 
28 
29 

708 
755 
802 

713 
759 
806 

717 
764 

811 

722 

769 
816 

727 

774 
820 

731 

778 

825 

736 

783 
830 

741 

788 
834 

745 

792 
839 

750 
797 
844 

7 
8 
9 

3.5 
4.0 
4.5 

2.8 
3.2 
3.6 

930 

848 

853 

858 

862 

867 

872 

876 

881 

886 

890 

31 
32 
33 

895 
942 

988 

900 
946 
993 

904 
951 
997 

90<) 

956 

*002 

914 

960 

*007 

918 

965 

*011 

923 

970 

*016 

928 

974 

*021 

932 

979 

*025 

937 

984 

*030 

a4 

35 
36 

97035 
081 
128 

039 
086 
132 

044 
090 
137 

049 
095 
142 

053 
100 
146 

058 
104 
151 

063 
109 
155 

067 
114 
160 

072 
118 
165 

077 
123 
169 

37 
38 
39 

174 

220 
267 

179 
225 

271 

183 
230 

276 

188 
234 

280 

192 
239 

285 

197 
243 
2SX) 

336 

202 
248 
294 

206 
253 
299 

211 
257 
304 

216 
262 
308 

940 

313 

317 

322 

327 

331 

340 

345 

350 

354 

41 
42 
43 

359 
405 
451 

364 
410 
456 

368 
414 
460 

373 
419 
465 

377 
424 
470 

382 
428 
474 

387 
433 
479 

391 
437 
483 

39(3 
442 

488 

400 
447 
493 

44 
45 
46 

497 
543 
589 

502 
548 
594 

506 
552 
598 

511 
557 
603 

516 
5()2 
607 

520 
56(5 
612 

525 
571 
617 

529 
575 
621 

534 
580 
626 

539 
585 
630 

47 

48 
49 

635 
681 

727 

(MO 
685 
731 

644 

690 
736 

649 
(595 
740 

653 

699 
745 

658 
704 
749 

663 

708 
754 

667 
713 
759 

804 

672 
717 
763 

676 
722 
768 

950 

772 

777 

782 

786 

791 

795 

800 

809 

813 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Prop.  Pts. 

Table  3.    Number  Logarithms 


195 


0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Prop.  Pts. 

950 

97  772 

777 

782 

786 

791 

795 

800 

804 

809 

813 

51 
52 
53 

818 
864 
909 

823 
868 
914 

827 
873 
918 

832 
877 
923 

836 
882 
928 

841 
886 
932 

845 
891 
937 

850 
896 
941 

855 
900 
946 

859 
905 
950 

54 
55 
56 

955 

98  000 

046 

959 
005 
050 

964 
009 
055 

f)68 
014 
059 

973 
019 
064 

978 
023 
068 

982 
028 
073 

987 
032 
078 

991 

037 
082 

996 
041 

087 

57 
58 
59 

091 
137 

182 

096 
141 
186 

100 
146 
191 

105 
150 
195 

109 
155 
200 

114 
159 
204 

118 
164 

209 

123 
1(58 
214 

127 
173 

218 

132 
177 
223 

960 

227 

232 

236 

241 

245 

250 

254 

259 

263 

268 

61 
62 
63 

272 
318 
363 

277 
322 
367 

281 
327 
372 

286 
331 
376 

290 
336 
381 

295 
340 
385 

299 
345 
390 

304 
349 
394 

30S 
354 
399 

313 
358 
403 

64 
65 
66 

408 
453 
498 

412 
457 
502 

417 
462 
507 

421 
466 
511 

426 
471 
516 

430 
475 
520 

435 
480 
525 

439 
484 
529 

444 
489 
534 

448 
493 
538 

67 
68 
69 

543 

588 
632 

547 
592 
637 

552 
597 
641 

556 
601 
646 

561 

605 
650 

565 
610 
655 

570 
614 

659 

574 
619 
664 

579 
623 

668 

583 
628 
673 

970 

677 

682 

686 

691 

695 

700 

704 

709 

713 

717 

71 
72 
73 

722 
767 
811 

726 
771 

816 

731 
776 

820 

735 

780 
825 

740 

784 
829 

744 
789 
834 

749 
793 

838 

753 
798 
843 

758 
802 
847 

762 
807 
851 

1 
2 

5 

0.5 
1.0 

4 

0.4 

0.8 

74 
75 
76 

856 
900 
945 

860 
905 
949 

865 
909 
954 

869 
914 
958 

874 
918 
963 

878 
923 
967 

883 
927 
972 

887 
932 
976 

892 
936 
981 

896 
941 
985 

3 
4 
5 
6 

1.5 
2.0 
2.5 
3  0 

1.2 
1.6 
2.0 
2.4 

77 
78 
79 

989 
99  034 

078 

994 
038 
083 

998 
043 

087 

*003 
047 
092 

*007 
052 
096 

*012 
056 
100 

*016 
0(a 
105 

*021 
065 
109 

*025 
069 
114 

*029 
074 

118 

7 
8 
9 

3.5 
4.0 
4.5 

2.8 
3.2 
3.6 

980 

123 

127 

131 

136 

140 

145 

149 

154 

158 

162 

81 
82 
83 

167 
211 
255 

171 
216 
260 

176 
220 
264 

180 
224 
269 

185 
229 
273 

189 
233 

277 

193 

238 
282 

198 
242 
286 

202 
247 
291 

207 
251 
295 

84 

85 
86 

300 
344 

388 

304 
348 
392 

308 
352 
396 

313 
357 
401 

317 
361 
405 

322 
366 
410 

326 
370 
414 

330 
374 
419 

335 
379 
423 

339 
383 
427 

87 
88 
89 

432 
476 

520 

436 
480 
524 

441 

484 

528 

445 

489 
533 

449 
493 
537 

454 

498 
542 

458 
502 
546 

463 

506 
550 

467 
511 

555 

471 

'515 

559 

990 

5fJ4 

568 

572 

577 

581 

585 

590 

5i>4 

599 

603 

91 
92 
93 

607 
651 
695 

612 
656 
699 

616 
660 
704 

621 

664 
708 

625 
669 
712 

629 
673 
717 

634 
677 
721 

638 
682 
726 

642 
686 
730 

647 
691 
734 

94 
95 
96 

739 

782 
826 

743 

787 
8.'^ 

747 
791 
835 

752 
795 
839 

756 
800 
843 

760 
804 
848 

765 

808 
852 

769 
813 

856 

774 

817 
861 

778 
822 
865 

97 
98 
99 

870 
913 
957 

874 
917 
%1 

878 
922 
965 

883 
926 
970 

887 
930 
974 

891 
935 
978 

896 
939 
983 

900 
944 

987 

904 
948 
991 

909 
952 
996 

1000 

00  000 

004 

009 

013 

017 

022 

026 

030 

035 

039 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Prop.  Pts. 

196 


Table  4.    Trigonometric  Logarithms 


0°  (180°) 

(359°. 

)  179° 

1   Sin   1 

Cos 

Tan 

Cot 

Sec 

Csc 

0 

— 

0.00  000 

— 

— 

0.00  000 

— 

60 

1 

6.46  373 

.00  000 

6.46  373 

3.53  627 

.00  000 

3.53  627 

59 

2 

6.76  476 

.00  000 

6.76  476 

3.23  524 

.00  000 

.23  524 

58 

3 

6.94  085 

.00  000 

6.94  085 

3.05  915 

.00  000 

.05  915 

57 

4 

7.06  579 

.00  000 

7.06  579 

2.93  421 

.00  000 

2.93  421 

56 

5 

7.16  270 

0.00  000 

7.16  270 

2.83  730 

0.00  000 

2.83  730 

55 

6 

.24  188 

.00  000 

.24  188 

.75  812 

.00  000 

.75  812 

54 

7 

.30  882 

.00  000 

.30  882 

.69  118 

.00  000 

.69  118 

53 

8 

.36  682 

.00  000 

.36  682 

.63  318 

.00  000 

.63  318 

52 

9 

.41  797 

.00  000 

.41  797 

.58  203 

.00  000 

.58  203 

51 

10 

7.46  373 

0.00  000 

7.46  373 

2.53  627 

0.00  000 

2.53  627 

50 

11 

.50  512 

.00  000 

.50  512 

.49  488 

.00  000 

.49  488 

49 

12 

.54  291 

.00  000 

.54  291 

.45  709 

.00  000 

.45  709 

48 

13 

.57  767 

.00  000 

.57  767 

.42  233 

.00  000 

.42  233 

47 

14 

.60  985 

.00  000 

.60  986 

.39  014 

.00  000 

..39  015 

46 

15 

7.63  982 

0.00  000 

7.63  982 

2.36  018 

0.00  000 

2.36  018 

45 

16 

.66  784 

.00  000 

.66  785 

.33  215 

.00  000 

.33  216 

44 

17 

.69  417 

9.99  999 

.69  418 

.30  582 

.00  001 

.30  583 

43 

18 

.71  900 

.99  999 

.71  900 

.28  100 

.00  001 

.28  100 

42 

19 

.74  248 

.99  999 

.74  248 

.25  752 

.00  001 

.25  752 

41 

20 

7.76  475 

9.99  999 

7.76  476 

2.23  524 

0.00  001 

2.23  525 

40 

21 

.78  594 

.99  999 

.78  595 

.21  405 

.00  001 

.21  406 

39 

22 

.80  615 

.99  999 

.80  615 

.19  385 

.00  001 

.19  385 

38 

23 

.82  545 

.99  999 

.82  546 

.17  454 

.00  001 

.17  455 

37 

24 

.84  393 

.99  999 

.84  394 

.15  606 

.00  001 

.15  607 

36 

25 

7.86  166 

9.99  999 

7.86  167 

2.13  833 

0.00  001 

2.13  834 

35 

26 

.87  870 

.99  999 

.87  871 

.12  129 

.00  001 

.12  1.30 

34 

27 

.89  509 

.99  999 

.89  510 

.10  490 

.00  001 

.10  491 

33 

28 

.91  088 

.99  999 

.91  089 

.08  911 

.00  001 

.08  912 

32 

29 

.92  612 

.99  998 

.92  613 

.07  387 

.00  002 

.07  388 

31 

30 

7.94  084 

9.99  998 

7.94  086 

2.05  914 

0.00  002 

2.05  916 

30 

31 

.95  508 

.99  998 

.95  510 

.04  490 

.00  002 

.04  492 

29 

32 

.96  887 

.99  998 

.96  889 

.03  111 

.00  002 

.03  113 

28 

33 

.98  223 

.99  998 

.98  225 

.01  775 

.00  002 

.01  777 

27 

34 

.99  520 

.99  998 

.99  522 

.00  478 

.00  002 

.00  480 

26 

35 

8.00  779 

9.99  998 

8.00  781 

1.99  219 

0.00  002 

1.99  221 

25 

36 

.02  002 

.99  998 

.02  004 

.97  996 

.00  002 

.97  998 

24 

37 

.03  192 

.99  997 

.03  194 

.96  806 

.00  003 

.96  808 

23 

38 

.04  350 

.99  997 

.04  353 

.95  647 

.00  003 

.95  650 

22 

39 

.05  478 

.99  997 

.05  481 

.94  519 

.00  003 

.94  522 

21 

40 

8.06  578 

9.99  997 

8.06  581 

1.93  419 

0.00  003 

1.93  422 

20 

41 

.07  650 

.99  997 

.07  653 

.92  347 

.00  003 

.92  350 

19 

42 

.08  696 

.99  997 

.08  700 

.91  300 

.00  003 

.91  304 

18 

43 

.09  718 

.99  997 

.09  722 

.90  278 

.00  003 

.90  282 

17 

44 

.10  717 

.99  996 

.10  720 

.89  280 

.00  004 

.89  283 

16 

45 

8.11  693 

9.99  996 

8.11  696 

1.88  304 

0.00  004 

1.88  307 

15 

46 

.12  647 

.99  996 

.12  651 

.87  349 

.00  004 

.87  353 

14 

47 

.13  581 

.99  996 

.13  585 

.86  415 

.00  004 

.86  419 

13 

48 

.14  495 

.99  996 

.14  500 

.85  500 

.00  004 

.85  505 

12 

49 

.15  391 

.99  996 

.15  395 

.84  605 

.00  004 

.84  609 

11 

50 

8.16  268 

9.99  995 

8.16  273 

1.83  727 

0.00  005 

1.83  732 

10 

51 

.17  128 

.99  995 

.17  133 

.82  867 

.00  005 

.82  872 

9 

52 

.17  971 

.99  995 

.17  976 

.82  024 

.00  005 

.82  029 

8 

53 

.18  798 

.99  995 

.18  804 

.81  196 

.00  005 

.81  202 

7 

54 

.19  610 

.99  995 

.19  616 

.80  384 

.00  005 

.80  390 

6 

55 

8.20  407 

9.99  994 

8.20  413 

1.79  587 

0.00  006 

1.79  593 

5 

56 

.21  189 

.99  994 

.21  195 

.78  805 

.00  006 

.78  811 

4 

57 

.21  958 

.99  994 

.21  964 

.78  036 

.00  006 

.78  042 

3 

58 

.22  713 

.99  994 

.22  720 

.77  280 

.00  006 

.77  287 

2 

59 

.23  456 

.99  994 

.23  462 

.76  538 

.00  006 

.76  544 

1 

60 

8.24  186 

9.99  993 

8.24  192 

1.75  808 

0.00  007 
Csc 

1.75  814 

0 

i   Cos 

Sin 

Cot 

Tan 

Sec 

/ 

90°  (270°) 


(269°)  89<= 


Table  4.    Trigonometric  Logarithms 


197 


1°  (181°) 

(358°)  178° 

' 

Sin 

Cos 

Tan 

Cot 

Sec 

Csc 

0 

8.24  186 

9.99  993 

8.24  192 

1.75  808 

0.00  007 

1.75  814 

60 

1 

.24  903 

.99  993 

.24  910 

.75  090 

.00  007 

.75  097 

59 

2 

.25  609 

.99  993 

.25  616 

.74  384 

.00  007 

.74  391 

58 

3 

.26  304 

.99  993 

.26  312 

.73  688 

.00  007 

.73  696 

57 

4 

.26  988 

.99  992 

.26  996 

.73  004 

.00  008 

.73  012 

56 

5 

8.27  661 

9.99  992 

8.27  669 

1.72  331 

0.00  008 

1.72  339 

55 

6 

.28  324 

.99  992 

.28  332 

.71  668 

.00  008 

.71  676 

54 

7 

.28  977 

.99  992 

.28  986 

.71  014 

.00  008 

.71  023 

53 

8 

.29  621 

.99  992 

.29  629 

.70  371 

.00  008 

.70  379 

52 

9 

.30  255 

.99  991 

.30  263 

.69  737 

.00  009 

.69  745 

51 

10 

8.30  879 

9.99  991 

8.30  888 

1.69  112 

0.00  009 

1.69  121 

50 

11 

.31  495 

.99  991 

.31  505 

.68  495 

.00  009 

.68  505 

49 

12 

.32  103 

.99  990 

.32  112 

.67  888 

.00  010 

.67  897 

48 

13 

.32  702 

.99  990 

.32  711 

.67  289 

.00  010 

.67  298 

47 

14 

.33  292 

.99  990 

.33  302 

.66  698 

.00  010 

.66  708 

46 

15 

8.33  875 

9.99  990 

8.33  886 

1.66  114 

0.00  010 

1.66  125 

45 

16 

.34  450 

.99  989 

.34  461 

.65  539 

00.011 

65  550 

44 

17 

.35  018 

.99  989 

.35  029 

.64  971 

.00  011 

.64  982 

43 

18 

.35  578 

.99  989 

.35  590 

.64  410 

.00  011 

.64  422 

42 

19 

.36  131 

.99  989 

.36  143 

.63  857 

.00  011 

.63  869 

41 

20 

8.36  678 

9.99  988 

8.36  689 

1.63  311 

0.00  012 

1.63  322 

40 

21 

.37  217 

.99  988 

.37  229 

.62  771 

.00  012 

.62  783 

39 

22 

.37  750 

.99  988 

.37  762 

.62  238 

.00  012 

.62  250 

38 

23 

.38  276 

.99  987 

.38  289 

.61  711 

.00  013 

.61  724 

37 

24 

.38  796 

.99  987 

.38  809 

.61  191 

.00  013 

.61  204 

36 

25 

8.39  310 

9.99  987 

8.39  323 

1.60  677 

0.00  013 

1.60  690 

35 

26 

.39  818 

.99  986 

.39  832 

.60  168 

.00  014 

.60  182 

34 

27 

.40  320 

.99  986 

.40  334 

.59  666 

.00  014 

.59  680 

33 

28 

.40  816 

.99  986 

.40  830 

.59  170 

.00  014 

.59  184 

32 

29 

.41  307 

.99  985 

.41  321 

.58  679 

.00  015 

.58  693 

31 

30 

8.41  792 

9.99  985 

8.41  807 

1.58  193 

0.00  015 

1.58  208 

30 

31 

.42  272 

.99  985 

.42  287 

.57  713 

.00  015 

.57  728 

29 

32 

.42  746 

.99  984 

.42  762 

.57  238 

.00  016 

.57  254 

28 

33 

.43  216 

.99  984 

.43  232 

.56  768 

.00  016 

.56  784 

27 

34 

.43  680 

.99  984 

.43  696 

.56  304 

.00  016 

.56  320 

26 

35 

8.44  139 

9.99  983 

8.44  156 

1.55  844 

0.00  017 

1.55  861 

25 

36 

.44  594 

.99  983 

.44  611 

.55  389 

.00  017 

.55  406 

24 

37 

.45  044 

.99  983 

.45  061 

.54  939 

.00  017 

.54  956 

23 

38 

.45  489 

.99  982 

.45  507 

.54  493 

.00  018 

.54  511 

22 

39 

.45  930 

.99  982 

.45  948 

.54  052 

.00  018 

.54  070 

21 

40 

8.46  366 

9.99  982 

8.46  385 

1.53  615 

0.00  018 

1.53  634 

20 

41 

.46  799 

.99  981 

.46  817 

.53  183 

.00  019 

.53  201 

19 

42 

.47  226 

.99  981 

.47  245 

.52  755 

.00  019 

.52  774 

18 

43  . 

.47  650 

.99  981 

.47  669 

.52  331 

.00  019 

.52  350 

17 

44 

.48  069 

.99  980 

.48  089 

.51911 

.00  020 

.51  931 

16 

45 

8.48  485 

9.99  980 

8.48  505 

1.51  495 

0.00  020 

1.51  515 

15 

46 

.48  896 

.99  979 

.48  917 

.51  083 

.00  021 

.51  104 

14 

47 

.49  304 

.99  979 

.49  325 

.50  675 

.00  021 

.50  696 

13 

48 

.49  708 

.99  979 

.49  729 

.50  271 

.00  021 

.50  292 

12 

49 

.50  108 

.99  978 

.50  130 

.49  870 

.00  022 

.49  892 

11 

50 

8.50  504 

9.99  978 

8.50  527 

1.49  473 

0.00  022 

1.49  496 

10 

51 

.50  897 

.99  977 

.50  920 

.49  080 

.00  023 

.49  103 

9 

52 

.51  287 

.99  977 

.51310 

.48  690 

.00  023 

.48  713 

8 

53 

.51  673 

.99  977 

.51  696 

.48  304 

.00  023 

.48  327 

7 

54 

.52  055 

.99  976 

.52  079 

.47  921 

.00  024 

.47  945 

6 

55 

8.52  434 

9.99  976 

8.52  459 

1.47  541 

0.00  024 

1.47  566 

5 

56 

.52  810 

.99  975 

.52  835 

.47  165 

.00  025 

.47  190 

4 

57 

.53  183 

.99  975 

.53  208 

.46  792 

.00  025 

.46  817 

3 

58 

.53  552 

.99  974 

.53  578 

.46  422 

.00  026 

.46  448 

2 

59 

.53  919 

.99  974 

.53  945 

.46  055 

.00  026 

.46  081 

1 

60 

8.54  282 

9.99  974 

8.54  308 

1.45  692 

0.00  026 

1.45  718 

0 

Cos 

Sin 

Cot 

Tan 

Csc 

Sec 

1 

91°  (271°) 


(268°)  88' 


198 


Table  4.    Trigonometric  Logarithms 


2°  (182°) 

(357°)  177° 

/ 

Sin 

Cos 

Tan 

Cot 

Sec 

Csc 

0 

8.54  282 

9.99  974 

8.54  308 

1.45  692 

0.00  026 

1.45  718 

60 

1 

.54  642 

.99  973 

.54  669 

.45  331 

.00  027 

.45  358 

59 

2 

.54  999 

.99  973 

.55  027 

.44  973 

.00  027 

.45  001 

58 

3 

.55  354 

.99  972 

.55  382 

.44  618 

.00  028 

.44  646 

57 

4 

.55  705 

.99  972 

.55  734 

.44  266 

.00  028 

.44  295 

56 

5 

8.56  054 

9.99  971 

8.56  083 

1.43  917 

0.00  029 

1.43  946 

55 

6 

.56  400 

.99  971 

.56  429 

.43  571 

.00  029 

.43  600 

54 

7 

.56  743 

.99  970 

.56  773 

.43  227 

.00  030 

.43  257 

53 

8 

.57  084 

.99  970 

.57  114 

.42  886 

.00  030 

.42  916 

52 

9 

.57  421 

.99  969 

.57  452 

.42  548 

.00  031 

.42  579 

51 

10 

8.57  757 

9.99  969 

8.57  788 

1.42  212 

0.00  031 

1.42  243 

50 

11 

.58  089 

.99  968 

.58  121 

.41  879 

.00  032 

.41911 

49 

12 

.58  419 

.99  968 

.58  451 

.41  549 

.00  032 

.41  581 

48 

13 

.58  747 

.99  967 

.58  779 

.41  221 

.00  033 

.41  253 

47 

14 

.59  072 

.99  967 

.59  105 

.40  895 

.00  033 

.40  928 

46 

15 

8.59  395 

9.99  967 

8.59  428 

1.40  572 

0.00  033 

1.40  605 

45 

16 

.59  715 

.99  966 

.59  749 

.40  251 

.00  034 

.40  285 

44 

17 

.60  033 

.99  966 

.60  068 

.39  932 

.00  034 

.39  967 

43 

18 

.60  349 

.99  965 

.60  384 

.39  616 

.00  035 

.39  651 

42 

19 

.60  662 

.99  964 

.60  698 

.39  302 

.00  036 

.39  338 

41 

20 

8.60  973 

9.99  964 

8.61  009 

1.38  991 

0.00  036 

1.39  027 

40 

21 

.61  282 

.99  963 

.61319 

.38  681 

.00  037 

.38  718 

39 

22 

.61  589 

.99  963 

.61  626 

.38  374 

.00  037 

.38  411 

38 

23 

.61  894 

.99  962 

.61  931 

.38  069 

.00  038 

.38  106 

37 

24 

.62  196 

.99  962 

.62  234 

.37  766 

.00  038 

.37  804 

36 

25 

8.62  497 

9.99  961 

8.62  535 

1.37  465 

0.00  039 

1.37  503 

35 

26 

.62  795 

.99  96.1 

.62  834 

.37  166 

.00  039 

.37  205 

34 

27 

.63  091 

.99  960 

.63  131 

.36  869 

.00  940 

.36  909 

33 

28 

.63  385 

.99  960 

.63  426 

.36  574 

.00  040 

.36  615 

32 

29 

.63  678 

.99  959 

.63  718 

.36  282 

.00  041 

.36  322 

31 

30 

8.63  968 

9.99  959 

8.64  009 

1.35  991 

0.00  041 

1.36  032 

30 

31 

.64  256 

.99  958 

.64  298 

.35  702 

.00  042 

.35  744 

29 

32 

.64  543 

.99  958 

.64  585 

.35  415 

.00  042 

.35  457 

28 

33 

.64  827 

.99  957 

.64  870 

.35  130 

.00  043 

.35  173 

27 

34 

.65  110 

.99  956 

.65  154 

.34  846 

.00  044 

.34  890 

26 

35 

8.65  391 

9.99  956 

8.65  435 

1.34  565 

0.00  044 

1.34  609 

25 

36 

.65  670 

.99  955 

.65  715 

.34  285 

.00  045 

.34  330 

24 

37 

.65  947 

.99  955 

.65  993 

.34  007 

.00  045 

.34  053 

23 

38 

.66  223 

.99  954 

.66  269 

.33  731 

.00  046 

.33  777 

22 

39 

.66  497 

.99  954 

.66  543 

.33  457 

.00  046 

.33  503 

21 

40 

8.66  769 

9.99  953 

8.66  816 

1.33  184 

0.00  047 

1.33  231 

20 

41 

.67  039 

.99  952 

.67  087 

.32  913 

.00  048 

.32  961 

19 

42 

.67  308 

.99  952 

.67  356 

.32  644 

.00  048 

.32  692 

18 

43 

.67  575 

.99  951 

.67  624 

.32  376 

.00  049 

.32  425 

17 

44 

.67  841 

.99  951 

.67  890 

.32  110 

.00  049 

.32  159 

16 

45 

8.68  104 

9.99  950 

8.68  154 

1.31  846 

0.00  050 

1.31  896 

15 

46 

.68  367 

.99  949 

.68  417 

.31  583 

.00  051 

.31  633 

14 

47 

.68  627 

.99  949 

.68  678 

.31  322 

.00  051 

.31  373 

13 

48 

.68  886 

.99  948 

.68  938 

.31  062 

.00  052 

.31  114 

12 

49 

.69  144 

.99  948 

.69  196 

.30  804 

.00  052 

.30  856 

11 

50 

8.69  400 

9.99  947 

8.69  453 

1.30  547 

0.00  053 

1.30  600 

10 

51 

.69  654 

.99  946 

.69  708 

.30  292 

.00  054 

.30  346 

9 

52 

.69  907 

.99  946 

.69  962 

.30  038 

.00  054 

.30  093 

8 

53 

.70  159 

.99  945 

.70  214 

.29  786 

.00  055 

.29  841 

7 

54 

.70  409 

.99  944 

.70  465 

.29  535 

.00  056 

.29  591 

6 

55 

8.70  658 

9.99  944 

8.70  714 

1.29  286 

0.00  056 

1.29  342 

5 

56 

.70  905 

.99  943 

.70  962 

.29  038 

.00  057 

.29  095 

4 

57 

.71  151 

.99  942 

.71  208 

.28  792 

.00  058 

.28  849 

3 

58 

.71  395 

.99  942 

.71  453 

.28  547 

.00  058 

.28  605 

2 

59 

.71  638 

.99  941 

.71  697 

.28  303 

.00  059 

.28  362 

1 

60 

8.71  880 

9.99  940 

8.71  940 

1.28  060 

0.00  060 

1.28  120 

0 

Cos 

Sin 

Cot   1 

1  Tan   I 

Csc 

Sec 

' 

92°  (272°) 


(267°)  87' 


Table  4.    Trigonometric  Logarithms 


199 


3°  (183°) 

(356°)  176° 

/ 

Sin 

Cos 

1   Tan 

Cot 

Sec 

Csc 

0 

8.71  880 

9.99  940 

8.71  940 

1.28  060 

0.00  060 

1.28  120 

60 

1 

.72  120 

.99  940 

.72  181 

.27  819 

.00  060 

.27  880 

59 

2 

.72  359 

.99  939 

.72  420 

.27  580 

.00  061 

.27  641 

58 

3 

.72  597 

.99  938 

.72  659 

.27  341 

.00  062 

.27  403 

57 

4 

.72  834 

.99  938 

.72  896 

.27  104 

.00  062 

.27  166 

56 

5 

8.73  069 

9.99  937 

8.73  132 

1.26  868 

0.00  063 

1.26  931 

55 

6 

.73  303 

.99  936 

.73  366 

.26  634 

.00  064 

.26  697 

54 

7 

.73  535 

.99  936 

.73  600 

.26  400 

.00  064 

.26  465 

53 

8 

.73  767 

.99  935 

.73  832 

.26  168 

.00  065 

.26  233 

52 

9 

.73  997 

.99  934 

.74  063 

.25  937 

.00  066 

.26  003 

51 

10 

8.74  226 

9.99  934 

8.74  292 

1.25  708 

0.00  066 

1.25  774 

50 

11 

.74  454 

.99  933 

.74  521 

.25  479 

.00  067 

.25  546 

49 

12 

.74  680 

.99  932 

.74  748 

.25  252 

.00  068 

.25  320 

48 

13 

.74  906 

.99  932 

.74  974 

.25  026 

.00  068 

.25  094 

47 

14 

.75  130 

.99  931 

.75  199 

.24  801 

.00  069 

.24  870 

46 

15 

8.75  353 

9.99  930 

8.75  423 

1.24  577 

0.00  070 

1.24  647 

45 

16 

.75  575 

.99  929 

.75  645 

.24  355 

.00  071 

.24  425 

44 

17 

.75  795 

.99  929 

.75  867 

.24  133 

.00  071 

.24  205 

43 

18 

.76  015 

.99  928 

.76  087 

.23  913 

.00  072 

.23  985 

42 

19 

.76  234 

.99  927 

.76  306 

.23  694 

.00  073 

.23  766 

41 

20 

8.76  451 

9.99  926 

8.76  525 

1.23  475 

0.00  074 

1.23  549 

40 

21 

.76  667 

.99  926 

.76  742 

.23  258 

.00  074 

.23  333 

39 

22 

.76  883 

.99  925 

.76  958 

.23  042 

.00  075 

.23  117 

38 

23 

.77  097 

.99  924 

.77  173 

.22  827 

.00  076 

.22  903 

37 

24 

.77  310 

.99  923 

.77  387 

.22  613 

.00  077 

.22  690 

36 

25 

8.77  522 

9.99  923 

8.77  600 

1.22  400 

0.00  077 

1.22  478 

35 

26 

.77  733 

.99  922 

.77  811 

.22  189 

.00  078 

.22  267 

34 

27 

.77  943 

.99  921 

.78  022 

.21  978 

.00  079 

.22  057 

33 

28 

.78  152 

.99  920 

.78  232 

.21  768 

.00  080 

.21  848 

32 

29 

.78  360 

.99  920 

.78  441 

.21  559 

.00  080 

.21  640 

31 

30 

8.78  568 

9.99  919 

8.78  649 

1.21  351 

0.00  081 

1.21  432 

30 

31 

.78  774 

.99  918 

.78  855 

.21  145 

.00  082 

.21  226 

29 

32 

.78  979 

.99  917 

.79  061 

.20  939 

.00  083 

.21  021 

28 

33 

.79  183 

.99  917 

.79  266 

.20  734 

.00  083 

.20  817 

27 

34 

.79  386 

.99  916 

.79  470 

.20  530 

.00  084 

.20  614 

26 

35 

8.79  588 

9.99  915 

8.79  673 

1.20  327 

0.00  085 

1.20  412 

25 

36 

.79  789 

.99  914 

.79  875 

.20  125 

.00  086 

.20  211 

24 

37 

.79  990 

.99  913 

.80  076 

.19  924 

.00  087 

.20  010 

23 

38 

.80  189 

.99  913 

.80  277 

.19  723 

.00  087 

.19  811 

22 

39 

.80  388 

.99  912 

.80  476 

.19  524 

.00  088 

.19  612 

21 

40 

8.80  585 

9.99  911 

8.80  674 

1.19  326 

0.00  089 

1.19  415 

20 

41 

.80  782 

.99  910 

.80  872 

.19  128 

.00  090 

.19  218 

19 

42 

.80  978 

.99  909 

.81  068 

.18  932 

.00  091 

.19  022 

18 

43 

.81  173 

.99  909 

.81  264 

.18  736 

.00  091 

.18  827 

17 

44 

.81  367 

.99  908 

.81  459 

.18  541 

.00  092 

.18  633 

16 

45 

8.81  560 

9.99  907 

8.81  653 

1.18  347 

0.00  093 

1.18  440 

15 

46 

.81  752 

.99  906 

.81  846 

.18  154 

.00  094 

.18  248 

14 

47 

.81  944 

.99  905 

.82  038 

.17  962 

.00  095 

.18  056 

13 

48 

.82  134 

.99  904 

.82  230 

.17  770 

.00  096 

.17  866 

12 

49 

.82  324 

.99  904 

.82  420 

.17  580 

.00  096 

.17  676 

11 

50 

8.82  513 

9.99  903 

8.82  610 

1.17  390 

0.00  097 

1.17  487 

10 

51 

.82  701 

.99  902 

.82  799 

.17  201 

.00  098 

.17  299 

9 

52 

.82  888 

.99  901 

.82  987 

.17  013 

.00  099 

.17  112 

8 

53 

.83  075 

.99  900 

.83  175 

.16  825 

.00  100 

.16  925 

7 

54 

.83  261 

.99  899 

.83  361 

.16  639 

.00  101 

.16  739 

6 

55 

8.83  446 

9.99  898 

8.83  547 

1.16  453 

0.00  102 

1.16  554 

5 

56 

.83  630 

.99  898 

.83  732 

.16  268 

.00  102 

.16  370 

4 

57 

.83  813 

.99  897 

.83  916 

.16  084 

.00  103 

.16  187 

3 

58 

.83  996 

.99  896 

.84  100 

.15  900 

.00  104 

.16  004 

2 

59 

.84  177 

.99  895 

.84  282 

.15  718 

.00  105 

.15  823 

1 

60 

8.84  358 

9.99  894 

8.84  464 

1.15  536 

0.00  106 

1.15  642 

0 

Cos 

Sin 

Cot   ! 

Tan 

Csc 

Sec 

/ 

93°  (273°) 


(266°)  86= 


200 


Table  4.    Trigonometric  Logarithms 


4°  (184°) 

(355°)  175° 

1 

Sin 

Cos 

Tan 

Cot 

Sec 

Csc 

0 

8.84  358 

9.99  894 

8.84  464 

1.15  536 

0.00  106 

1.15  642 

60 

1 

.84  539 

.99  893 

.84  646 

.15  354 

.00  107 

.15  461 

59 

2 

.84  718 

.99  892 

.84  826 

.15  174 

.00  108 

.15  282 

58 

3 

.84  897 

.99  891 

.85  006 

.14  994 

.00  109 

.15  103 

57 

4 

.85  075 

.99  891 

.85  185 

.14  815 

.00  109 

.14  925 

56 

5 

8.85  252 

9.99  890 

8.85  363 

1.14  637 

0.00  110 

1.14  748 

55 

6 

.85  429 

.99  889 

.85  540 

.14  460 

.00  111 

.14  571 

54 

7 

.85  605 

.99  888 

.85  717 

.14  283 

.00  112 

.14  395 

53 

8 

.85  780 

.99  887 

.85  893 

.14  107 

.00  113 

.14  220 

52 

9 

.85  955 

.99  886 

.86  069 

.13  931 

.00  114 

.14  045 

51 

10 

8.86  128 

9.99  885 

8.86  243 

1.13  757 

0.00  115 

1.13  872 

50 

11 

.86  301 

.99  884 

.86  417 

.13  583 

.00  116 

.13  699 

49 

12 

.86  474 

.99  883 

.86  591 

.13  409 

.00  117 

.13  526 

48 

13 

.86  645 

.99  882 

.86  763 

.13  237 

.00  118 

.13  355 

47 

14 

.86  816 

.99  881 

.86  935 

.13  065 

.00  119 

.13  184 

46 

15 

8.86  987 

9.99  880 

8.87  106 

1.12  894 

0.00  120 

1.13  013 

45 

16 

.87  156 

.99  879 

.87  277 

.12  723 

.00  121 

.12  844 

44 

17 

.87  325 

.99  879 

.87  447 

.12  553 

.00  121 

.12  675 

43 

18 

.87  494 

.99  878 

.87  616 

.12  384 

.00  122 

.12  506 

42 

19 

.87  661 

.99  877 

.87  785 

.12  215 

.00  123 

.12  339 

41 

20 

8.87  829 

9.99  876 

8.87  953 

1.12  047 

0.00  124 

1.12  171 

40 

21 

.87  995 

.99  875 

.88  120 

.11  880 

.00  125 

.12  005 

39 

22 

.88  161 

.99  874 

.88  287 

.11  713 

00  126 

.11  839 

38 

23 

.88  326 

.99  873 

.88  453 

.11  547 

.00  127 

.11  674 

37 

24 

.88  490 

.99  872 

.88  618 

.11  382 

.00  128 

.11  510 

36 

25 

8.88  654 

9.99  871 

8.88  783 

1.11  217 

0.00  129 

1.11  346 

35 

26 

.88  817 

.99  870 

.88  948 

.11  052 

.00  130 

.11  183 

34 

27 

.88  980 

.99  869 

.89  111 

.10  889 

.00  131 

.11  020 

33 

28 

.89  142 

.99  868 

.89  274 

.10  726 

.00  132 

.10  858 

32 

29 

.89  304 

.99  867 

.89  437 

.10  563 

.00  133 

.10  696 

31 

30 

8.89  464 

9.99  866 

8.89  598 

1.10  402 

0.00  134 

1.10  536 

30 

31 

.89  625 

.99  865 

.89  760 

.10  240 

.00  135 

.10  375 

29 

32 

.89  784 

.99  864 

.89  920 

.10  080 

.00  136 

.10  216 

28 

33 

.89  943 

.99  863 

.90  080 

.09  920 

.00  137 

.10  057 

27 

34 

.90  102 

.99  862 

.90  240 

.09  760 

.00  138 

.09  898 

26 

35 

8.90  260 

9.99  861 

8.90  399 

1.09  601 

0.00  139 

1.09  740 

25 

36 

.90  417 

.99  860 

.90  557 

.09  443 

.00  140 

.09  583 

24 

37 

.90  574 

.99  859 

.90  715 

.09  285 

.00  141 

.09  426 

23 

38 

.90  730 

.99  858 

.90  872 

.09  128 

.00  142 

.09  270 

22 

39 

.90  885 

.99  857 

.91  029 

.08  971 

.00  143 

.09  115 

21 

40 

8.91  040 

9.99  856 

8.91  185 

1.08  815 

0.00  144 

1.08  960 

20 

41 

.91  195 

.99  855 

.91  340 

.08  660 

.00  145 

.08  805 

19 

42 

.91  349 

.99  854 

.91  495 

.08  505 

.00  146 

.08  651 

18 

43 

.91  502 

.99  853 

.91  650 

.08  350 

.00  147 

.08  498 

17 

44 

.91  655 

.99  852 

.91  803 

.08  197 

.00  148 

.08  345 

16 

45 

8.91  807 

9.99  851 

8.91  957 

1.08  043 

0.00  149 

1.08  193 

15 

46 

.91  959 

.99  850 

.92  110 

.07  890 

.00  150 

.08  041 

14 

47 

.92  110 

.99  848 

.92  262 

.07  738 

.00  152 

.07  890 

13 

48 

.92  261 

.99  847 

.92  414 

.07  586 

.00  153 

.07  739 

12 

49 

.92  411 

.99  846 

.92  565 

.07  435 

.00  154 

.07  589 

11 

50 

8.92  561 

9.99  845 

8.92  716 

1.07  284 

0.00  155 

1.07  439 

10 

51 

.92  710 

.99  844 

.92  866 

.07  134 

.00  156 

.07  290 

9 

52 

.92  859 

.99  843 

.93  016 

.06  984 

.00  157 

.07  141 

8 

53 

.93  007 

.99  842 

.93  165 

.06  835 

.00  158 

.06  993 

7 

54 

.93  154 

.99  841 

.93  313 

.06  687 

.00  159 

.06  846 

6 

55 

8.93  301 

9.99  840 

8.93  462 

1.06  538 

0.00  160 

1.06  699 

5 

56 

.93  448 

.99  839 

.93  609 

.06  391 

.00  161 

.06  552 

4 

57 

.93  594 

.99  838 

.93  756 

.06  244 

.00  162 

.06  406 

3 

58 

.93  740 

.99  837 

.93  903 

.06  097 

.00  163 

.06  260 

2 

59 

.93  885 

.99  836 

.94  049 

.05  951 

.00  164 

.06  115 

1 

60 

8.94  030 

9.99  834 

8.94  195 

1.05  805 

0.00  166 

1.05  970 

0 

Cos 

Sin 

Cot 

Tan 

Csc 

Sec 

/ 

94°  (274°) 


(265°)  ^h' 


Table  4.    Trigonometric  Logarithms 


201 


5°  (185°) 

(354°)  174° 

/ 

Sin 

Cos 

Tan 

Cot 

Sec 

Csc 

0 

8.94  030 

9.99  834 

8.94  195 

1.05  805 

0.00  166 

1.05  970 

60 

1 

.94  174 

.99  833 

.94  340 

.05  660 

.00  167 

.05  826 

59 

2 

.94  317 

.99  832 

.94  485 

.05  515 

.00  168 

.05  683 

58 

3 

.94  461 

.99  831 

.94  630 

.05  370 

.00  169 

.05  539 

57 

4 

.94  603 

.99  830 

.94  773 

.05  227 

.00  170 

.05  397 

56 

5 

8.94  746 

9.99  829 

8.94  917 

1.05  083 

0.00  171 

1.05  254 

55 

6 

.94  887 

.99  828 

.95  060 

.04  940 

.00  172 

.05  113 

54 

7 

.95  029 

.99  827 

.95  202 

.04  798 

.00  173 

.04  971 

53 

8 

.95  170 

.99  825 

.95  344 

.04  656 

.00  175 

.04  830 

52 

9 

.95  310 

.99  824 

.95  486 

.04  514 

.00  176 

.04  690 

51 

10 

8.95  450 

9.99  823 

8.95  627 

1.04  373 

0.00  177 

1.04  550 

50 

11 

.95  589 

.99  822 

.95  767 

.04  233 

.00  178 

.04  411 

49 

12 

.95  728 

.99  821 

.95  908 

.04  092 

.00  179 

.04  272 

48 

13 

.95  867 

.99  820 

.96  047 

.03  953 

.00  180 

.04  133 

47 

14 

.96  005 

.99  819 

.96  187 

.03  813 

.00  181 

.03  995 

46 

15 

8.96  143 

9.99  817 

8.96  325 

1.03  675 

0.00  183 

1.03  857 

45 

16 

.96  280 

.99  816 

.96  464 

.03  536 

.00  184 

.03  720 

44 

17 

.96  417 

.99815 

.96  602 

.03  398 

.00  185 

.03  583 

43 

18 

.96  553 

.99  814 

.96  739 

.03  261 

.00  186 

.03  447 

42 

19 

.96  689 

.99  813 

.96  877 

.03  123 

.00  187 

.03  311 

41 

20 

8.96  825 

9.99  812 

8.97  013 

1.02  987 

0.00  188 

1.03  175 

40 

21 

.96  960 

.99  810 

.97  150 

.02  850 

.00  190 

.03  040 

39 

22 

.97  095 

.99  809 

.97  285 

.02  715 

.00  191 

.02  905 

38 

23 

.97  229 

.99  808 

.97  421 

.02  579 

.00  192 

.02  771 

37 

24 

.97  363 

.99  807 

.97  556 

.02  444 

.00  193 

.02  637 

36 

25 

8.97  496 

9.99  806 

8.97  691 

1.02  309 

0.00  194 

1.02  504 

35 

26 

.97  629 

.99  804 

.97  825 

.02  175 

.00  196 

.02  371 

34 

27 

.97  762 

.99  803 

.97  959 

.02  041 

.00  197 

.02  238 

33 

28 

.97  894 

.99  802 

.98  092 

.01  908 

.00  198 

.02  106 

32 

29 

.98  026 

.99  801 

.98  225 

.01  775 

.00  199 

.01  974 

31 

30 

8.98  157 

9.99  800 

8.98  358 

1.01  642 

0.00  200 

1.01  843 

30 

31 

.98  288 

.99  798 

.98  490 

.01  510 

.00  202 

.01  712 

29 

32 

.98  419 

.99  797 

.98  622 

.01  378 

.00  203 

.01  581 

28 

33 

.98  549 

.99  796 

.98  753 

.01  247 

.00  204 

.01  451 

27 

34 

.98  679 

.99  795 

.98  884 

.01  116 

.00  205 

.01  321 

26 

35 

8.98  808 

9.99  793 

8.99  015 

1.00  985 

0.00  207 

1.01  192 

25 

36 

.98  937 

.99  792 

.99  145 

.00  855 

.00  208 

.01  063 

24 

37 

.99  066 

.99  791 

.99  275 

.00  725 

.00  209 

.00  934 

23 

38 

.99  194 

.99  790 

.99  405 

.00  595 

.00  210 

.00  806 

22 

39 

.99  322 

.99  788 

.99  534 

.00  466 

.00  212 

.00  678 

21 

40 

8.99  450 

9.99  787 

8.99  662 

1.00  338 

0.00  213 

1.00  550 

20 

41 

.99  577 

.99  786 

.99  791 

.00  209 

.00  214 

.00  423 

19 

42 

.99  704 

.99  785 

.99  919 

.00  081 

.00  215 

.00  296 

18 

43 

.99  830 

.99  783 

9.00  046 

0.99  954 

.00  217 

.00  170 

17 

44 

.99  956 

.99  782 

.00  174 

.99  826 

.00  218 

.00  044 

16 

45 

9.00  082 

9.99  781 

9.00  301 

0.99  699 

0.00  219 

0.99  918 

15 

46 

.00  207 

.99  780 

.00  427 

.99  573 

.00  220 

.99  793 

14 

47 

.00  332 

.99  778 

.00  553 

.99  447 

.00  222 

.99  668 

13 

48 

.00  456 

.99  777 

.00  679 

.99  321 

.00  223 

.99  544 

12 

49 

.00  581 

.99  776 

.00  805 

.99  195 

.00  224 

.99  419 

11 

50 

9.00  704 

9.99  775 

9.00  930 

0.99  070 

0.00  225 

0.99  296 

10 

51 

.00  828 

.99  773 

.01  055 

.98  945 

.00  227 

.99  172 

9 

52 

.00  951 

.99  772 

.01  179 

.98  821 

.00  228 

.99  049 

8 

53 

.01  074 

.99  771 

.01  303 

.98  697 

.00  229 

.98  926 

7 

54 

.01  196 

.99  769 

.01  427 

.98  573 

.00  231 

.98  804 

6 

55 

9.01  318 

9.99  768 

9.01  550 

0.98  450 

0.00  232 

0.98  682 

5 

56 

.01  440 

.99  767 

.01  673 

.98  327 

.00  233 

.98  560 

4 

57 

.01  561 

.99  765 

.01  796 

.98  204 

.00  235 

.98  439 

3 

58 

.01  682 

.99  764 

.01  918 

.98  082 

.00  236 

.98  318 

2 

59 

.01  803 

.99  763 

.02  040 

.97  960 

.00  237 

.98  197 

1 

60 

9.01  923 

9.99  761 

9.02  162 
Cot 

0.97  838 

0.00  239 

0.98  077 

0 

Cos 

Sin 

Tan 

Csc 

Sec 

/ 

95°  (275°) 


(264°)  84° 


202 


Table  4.    Trigonometric  Logarithms 


6°  (186°) 

(353°)  173° 

/ 

Sin 

Cos 

Tan 

Cot 

Sec 

Csc 

0 

9.01  923 

9.99  761 

9.02  162 

0.97  838 

0.00  239 

0.98  077 

60 

1 

.02  043 

.99  760 

.02  283 

.97  717 

00  240 

.97  957 

59 

2 

.02  163 

.99  759 

.02  404 

.97  596 

.00  241 

.97  837 

58 

3 

.02  283 

.99  757 

.02  525 

.97  475 

.00  243 

.97  717 

57 

4 

.02  402 

.99  756 

.02  645 

.97  355 

.00  244 

.97  598 

56 

5 

9.02  520 

9.99  755 

9.02  766 

0.97  234 

0.00  245 

0.97  480 

55 

6 

.02  639 

.99  753 

.02  885 

.97  115 

.00  247 

.97  361 

54 

7 

.02  757 

.99  752 

.03  005 

.96  995 

.00  248 

.97  243 

53 

8 

.02  874 

.99  751 

.03  124 

.96  876 

.00  249 

.97  126 

52 

9 

.02  992 

.99  749 

.03  242 

.96  758 

.00  251 

.97  008 

51 

10 

9.03  109 

9.99  748 

9.03  361 

0.96  639 

0.00  252 

0.96  891 

50 

11 

.03  226 

.99  747 

.03  479 

.96  521 

.00  253 

.96  774 

49 

12 

.03  342 

.99  745 

.03  597 

.96  403 

.00  255 

.96  658 

48 

13 

.03  458 

.99  744 

.03  714 

.96  286 

.00  256 

.96  542 

47 

14 

.03  574 

.99  742 

.03  832 

.96  168 

.00  258 

.96  426 

46 

15 

9.03  690 

9.99  741 

9.03  948 

0.96  052 

0.00  259 

0.96  310 

45 

16 

.03  805 

.99  740 

.04  065 

.95  935 

.00  260 

.96  195 

44 

17 

.03  920 

.99  738 

.04  181 

.95  819 

.00  262 

.96  080 

43 

18 

.04  034 

.99  737 

.04  297 

.95  703 

.00  263 

.95  966 

42 

19 

.04  149 

.99  736 

.04  413 

.95  587 

.00  264 

.95  851 

41 

20 

9.04  262 

9.99  734 

9.04  528 

0.95  472 

0.00  266 

0.95  738 

40 

21 

.04  376 

.99  733 

.04  643 

.95  357 

.00  267 

.95  624 

39 

22 

.04  490 

.99  731 

.04  758 

.95  242 

.00  269 

.95  510 

38 

23 

.04  603 

.99  730 

.04  873 

.95  127 

.00  270 

.95  397 

37 

24 

.04  715 

.99  728 

.04  987 

.95  013 

.00  272 

.95  285 

36 

25 

9.04  828 

9.99  727 

9.05  101 

0.94  899 

0.00  273 

0.95  172 

35 

26 

.04  940 

.99  726 

.05  214 

.94  786 

.00  274 

.95  060 

34 

27 

.05  052 

.99  724 

.05  328 

.94  672 

.00  276 

.94  948 

33 

28 

.05  164 

.99  723 

.05  441 

.  94  559 

.00  277 

.94  836 

32 

29 

.05  275 

.99  721 

.05  553 

.94  447 

.00  279 

.94  725 

31 

30 

9.05  386 

9.99  720 

9.05  666 

0.94  334 

0.00  280 

0.94  614 

30 

31 

.05  497 

.99  718 

.05  778 

.94  222 

.00  282 

.94  503 

29 

32 

.05  607 

.99  717 

.05  890 

.94  110 

.00  283 

.94  393 

28 

33 

.05  717 

.99  716 

.06  002 

.93  998 

.00  284 

.94  283 

27 

34 

.05  827 

.99  714 

.06  113 

.93  887 

.00  286 

.94  173 

26 

35 

9.05  937 

9.99  713 

9.06  224 

0.93  776 

0.00  287 

0.94  063 

25 

36 

.06  046 

.99  711 

.06  335 

.93  665 

.00  289 

.93  954 

24 

37 

.06  155 

.99  710 

.06  445 

.93  555 

.00  290 

.93  845 

23 

38 

.06  264 

.99  708 

.06  556 

.93  444 

.00  292 

.93  736 

22 

39 

.06  372 

.99  707 

.06  666 

.93  334 

.00  293 

.93  628 

21 

40 

9.06  481 

9.99  705 

9.06  775 

0.93  225 

0.00  295 

0.93  519 

20 

41 

.06  589 

.99  704 

.06  885 

.93  115 

.00  296 

.93  411 

19 

42 

.06  696 

.99  702 

.06  994 

.93  006 

.00  298 

.93  304 

18 

43 

.06  804 

.99  701 

.07  103 

.92  897 

.00  299 

.93  196 

17 

44 

.06  911 

.99  699 

.07  211 

.92  789 

.00  301 

.93  089 

16 

45 

9.07  018 

9.99  698 

9.07  320 

0.92  680 

0.00  302 

0.92  982 

15 

46 

.07  124 

.99  696 

-.07  428 

.92  572 

.00  304 

.92  876 

14 

47 

.07  231 

.99  695 

.07  536 

.92  464 

.00  305 

.92  769 

13 

48 

.07  337 

.99  693 

.07  643 

.92  357 

00.307 

.92  663 

12 

49 

.07  442 

.99  692 

.07  751 

.92  249 

.00  308 

.92  558 

11 

50 

9.07  548 

9.99  690 

9.07  858 

0.92  142 

0.00  310 

0.92  452 

10 

51 

.07  653 

.99  689 

.07  964 

.92  036 

.00  311 

.92  347 

9 

52 

.07  758 

.99  687 

.08  071 

.91  929 

.00  313 

.92  242 

8 

53 

.07  863 

.99  686 

.08  177 

.91  823 

.00  314 

.92  137 

7 

54 

.07  968 

.99  684 

.08  283 

.91  717 

.00  316 

.92  032 

6 

55 

9.08  072 

9.99  683 

9.08  389 

0.91  611 

0.00  317 

0.91  928 

5 

56 

.08  176 

.99  681 

.08  495 

.91  505 

.00  319 

.91  824 

4 

57 

.08  280 

.99  680 

.08  600 

.91  400 

.00  320 

.91  720 

3 

58 

.08  383 

.99  678 

.08  705 

.91  295 

.00  322 

.91  617 

2 

59 

.08  486 

.99  677 

.08  810 

.91  190 

.00  323 

.91  514 

1 

60 

9.08  589 

9.99  675 

9.08  914 

0.91  086 

0.00  325 

0.91411 

0 

Cos 

Sin 

Qot 

Tan 

Csc 

Sec 

1 

96°  (276°) 


(263°)  83= 


Table  4.    Trigonometric  Logarithms 


203 


7°  (187°) 

(352° 

)  172° 

/ 

Sin 

Cos 

Tan 

Cot 

Sec 

Csc 

60 

0 

9.08  589 

9.99  675 

9.08  914 

0.91  086 

0.00  325 

0.91  411 

1 

.08  692 

.99  674 

.09  019 

.90  981 

.00  326 

.91  308 

59 

2 

.08  795 

.99  672 

.09  123 

.90  877 

.00  328 

.91  205 

58 

3 

.08  897 

.99  670 

.09  227 

.90  773 

.00  330 

.91  103 

57 

4 

.08  999 

.99  669 

.09  330 

.90  670 

.00  331 

.91  001 

56 

5 

9.09  101 

9.99  667 

9.09  434 

0.90  566 

0.00  333 

0.90  899 

55 

6 

.09  202 

.99  666 

.09  537 

.90  463 

.00  334 

.90  798 

54 

7 

.09  304 

.99  664 

.09  640 

.90  360 

.00  336 

.90  696 

53 

8 

.09  405 

.99  663 

.09  742 

.90  258 

.00  337 

.90  595 

52 

9 

.09  506 

.99  661 

.09  845 

.90  155 

.00  339 

.90  494 

51 

10 

9.09  606 

9.99  659 

9.09  947 

0.90  053 

0.00  341 

0.90  394 

50 

11 

.09  707 

.99  658 

.10  049 

.89  951 

.00  342 

.90  293 

49 

12 

.09  807 

.99  656 

.10  150 

.89  850 

.00  344 

.90  193 

48 

13 

.09  907 

.99  655 

.10  252 

.89  748 

.00  345 

.90  093 

47 

14 

.10  006 

.99  653 

.10  353 

.89  647 

.00  347 

.89  994 

46 

15 

9.10  106 

9.99  651 

9.10  454 

0.89  546 

0.00  349 

0.89  894 

45 

16 

.10  205 

.99  650 

.10  555 

.89  445 

.00  350 

.89  795 

44 

17 

.10  304 

.99  648 

.10  656 

.89  344 

.00  352 

.89  696 

43 

18 

.10  402 

.99  647 

.10  756 

.89  244 

.00  353 

.89  598 

42 

19 

.10  501 

.99  645 

.10  856 

.89  144 

.00  355 

.89  499 

41 

20 

9.10  599 

9.99  643 

9.10  956 

0.89  044 

0.00  357 

0.89  401 

40 

21 

.10  697 

.99  642 

.11  056 

.88  944 

.00  358 

.89  303 

39 

22 

.10  795 

.99  640 

.11  155 

.88  845 

.00  360 

.89  205 

38 

23 

.10  893 

.99  638 

.11  254 

.88  746 

.00  362 

.89  107 

37 

24 

.10  990 

.99  637 

.11  353 

.88  647 

.00  363 

.89  010 

36 

25 

9.11  087 

9.99  635 

9.11452 

0.88  548 

0.00  365 

0.88  913 

35 

26 

.11  184 

.99  633 

.11  551 

.88  449 

.00  367 

.88  816 

34 

27 

.11  281 

.99  632 

.11  649 

.88  351 

.00  368 

.88  719 

33 

28 

.11  377 

.99  630 

.11  747 

.88  253 

.00  370 

.88  623 

32 

29 

.11  474 

.99  629 

.11  845 

.88  155 

.00  371 

.88  526 

31 

30 

9.11  570 

9.99  627 

9.11  943 

0.88  057 

0.00  373 

0.88  430 

30 

31 

.11  666 

.99  625 

.12  040 

.87  960 

.00  375 

.88  334 

29 

32 

.11  761 

.99  624 

.12  138 

.87  862 

.00  376 

.88  239 

28 

33 

.11  857 

.99  622 

.12  235 

.87  765 

.00  378 

.88  143 

27 

34 

.11  952 

.99  620 

.12  332 

.87  668 

.00  380 

.88  048 

26 

35 

9.12  047 

9.99  618 

9.12  428 

0.87  572 

0.00  382 

0.87  953 

25 

36 

.12  142 

.99  617 

.12  525 

.87  475 

.00  383 

.87  858 

24 

37 

.12  236 

.99  615 

.12  621 

.87  379 

.00  385 

.87  764 

23 

38 

.12  331 

.99  613 

.12  717 

.87  283 

.00  387 

.87  669 

22 

39 

.12  425 

.99  612 

.12  813 

.87  187 

.00  388 

.87  575 

21 

40 

9.12  519 

9.99  610 

9.12  909 

0.87  091 

0.00  390 

0.87  481 

20 

41 

.12  612 

.99  608 

.13  004 

.86  996 

.00  392 

.87  388 

19 

42 

.12  706 

.99  607 

.13  099 

.86  901 

.00  393 

.87  294 

18 

43 

.12  799 

.99  605 

.13  194 

.86  806 

.00  395 

.87  201 

17 

44 

.12  892 

.99  603 

.13  289 

.86  711 

.00  397 

.87  108 

16 

45 

9.12  985 

9.99  601 

9.13  384 

0.86  616 

0.00  399 

0.87  015 

15 

46 

.13  078 

.99  600 

.13  478 

.86  522 

.00  400 

.86  922 

14 

47 

.13  171 

.99  598 

.13  573 

.86  427 

.00  402 

.86  829 

13 

48 

.13  263 

.99  596 

.13  667 

.86  333 

.00  404 

.86  737 

12 

49 

.13  355 

.99  595 

.13  761 

.86  239 

.00  405 

.86  645 

11 

50 

9.13  447 

9.99  593 

9.13  854 

0.86  146 

0.00  407 

0.86  553 

10 

51 

.13  539 

.99  591 

.13  948 

.86  052 

.00  409 

.86  461 

9 

52 

.13  630 

.99  589 

.14  041 

.85  959 

.00  411 

.86  370 

8 

53 

.13  722 

.99  588 

.14  134 

.85  866 

.00  412 

.86  278 

7 

54 

.13  813 

.99  586 

.14  227 

.85  773 

.00  414 

.86  187 

6 

55 

9.13  904 

9.99  584 

9.14  320 

0.85  680 

0.00  416 

0.86  096 

5 

56 

.13  994 

.99  582 

.14  412 

.85  588 

.00  418 

.86  006 

4 

57 

.14  085 

.99  581 

.14  504 

.85  496 

.00  419 

.85  915 

3 

58 

.14  175 

.99  579 

.14  597 

.85  403 

.00  421 

.85  825 

2 

59 

.14  266 

.99  577 

.14  688 

.85  312 

.00  423 

.85  734 

1 

60 

9.14  356 

9.99  575 

9.14  780 

0.85  220 

0.00  425 

0.85  644 

0 

Cos 

Sin 

Cot 

Tan 

Csc 

Sec 

/ 

97°  (277°) 


(262°)  82= 


204 


Table  4.    Trigonometric  Logarithms 


8°  (188°) 

(351°) 

171° 

' 

Sin 

Cos 

Tan 

Cot 

Sec 

Csc 

0 

9.14  356 

9.99  575 

9.14  780 

0.85  220 

0.00  425 

0.85  644 

60 

1 

.14  445 

.99  574 

.14  872 

.85  128 

.00  426 

.85  555 

59 

2 

.14  535 

.99  572 

.14  963 

.85  037 

.00  428 

.85  465 

58 

3 

.14  624 

.99  570 

.15  054 

.84  946 

.00  430 

.85  376 

57 

4 

.14  714 

.99  568 

.15  145 

.84  855 

.00  432 

.85  286 

56 

5 

9.14  803 

9.99  566 

9.15  236 

0.84  764 

0.00  434 

0.85  197 

55 

6 

.14  891 

.99  565 

.15  327 

.84  673 

.00  435 

.85  109 

54 

7 

.14  980 

.99  563 

.15  417 

.84  583 

.00  437 

.85  020 

53 

8 

.15  069 

.99  561 

.15  508 

.84  492 

.00  439 

.84  931 

52 

9 

.15  157 

.99  559 

.15  598 

.84  402 

.00  441 

.84  843 

51 

10 

9.15  245 

9.99  557 

9.15  688 

0.84  312 

0.00  443 

0.84  755 

50 

11 

.15  333 

.99  556 

.15  777 

.84  223 

.00  444 

.84  667 

49 

12 

.15  421 

.99  554 

.15  867 

.84  133 

.00  446 

.84  579 

48 

13 

.15  508 

.99  552 

.15  956 

.84  044 

.00  448 

.84  492 

47 

14 

.15  596 

.99  550 

.16  046 

.83  954 

.00  450 

.84  404 

46 

15 

9.15  683 

9.99  548 

9.16  135 

0.83  865 

0.00  452 

0.84  317 

45 

16 

.15  770 

.99  546 

.16  224 

.83  776 

.00  454 

.84  230 

44 

17 

.15  857 

.99  545 

.16  312 

.83  688 

.00  455 

.84  143 

43 

18 

.15  944 

.99  543 

.16  401 

.83  599 

.00  457 

.84  056 

42 

19 

.16  030 

.99  541 

.16  489 

.83  511 

.00  459 

.83  970 

41 

20 

9.16  116 

9.99  539 

9.16  577 

0.83  423 

0.00  461 

0.83  884 

40 

21 

.16  203 

.99  537 

.16  665 

.83  335 

.00  463 

.83  797 

39 

22 

.16  289 

.99  535 

.16  753 

.83  247 

.00  465 

.83  711 

38 

23 

.16  374 

.99  533 

.16  841 

.83  159 

.00  467 

.83  626 

37 

24 

.16  460 

.99  532 

.16  928 

.83  072 

.00  468 

.83  540 

36 

25 

9.16  545 

9.99  530 

9.17  016 

0.82  984 

0.00  470 

0.83  455 

35 

26 

.16  631 

.99  528 

.17  103 

.82  897 

.00  472 

.83  369 

34 

27 

.16  716 

.99  526 

.17  190 

.82  810 

.00  474 

.83  284 

33 

28 

.16  801 

.99  524 

.17  277 

.82  723 

.00  476 

.83  199 

32 

29 

.16  886 

.99  522 

.17  363 

.82  637 

.00  478 

.83  114 

31 

30 

9.16  970 

9.99  520 

9.17  450 

0.82  550 

0.00  480 

0.83  030 

30 

31 

.17  055 

.99  518 

.17  536 

.82  464 

.00  482 

.82  945 

29 

32 

.17  139 

.99  517 

.17  622 

.82  378 

.00  483 

.82  861 

28 

33 

.17  223 

.99  515 

.17  708 

.82  292 

.00  485 

.82  777 

27 

34 

.17  307 

.99  513 

.17  794 

.82  206 

.00  487 

.82  693 

26 

35 

9.17  391 

9.99  511 

9.17  880 

0.82  120 

0.00  489 

0.82  609 

25 

36 

.17  474 

.99  509 

.17  965 

.82  035 

.00  491 

.82  526 

24 

37 

.17  558 

.99  507 

.18  051 

.81  949 

.00  493 

.82  442 

23 

38 

.17  641 

.99  505 

.18  136 

.81  864 

.00  495 

.82  359 

22 

39 

.17  724 

.99  503 

.18  221 

.81  779 

.00  497 

.82  276 

21 

40 

9.17  807 

9.99  50-1 

9.18  306 

0.81  694 

0.00  499 

0.82  193 

20 

41 

.17  890 

.99  499 

.18  391 

.81  609 

.00  501 

.82  110 

19 

42 

.17  973 

.99  497 

.18  475 

.81  525 

.00  503 

.82  027 

18 

43 

.18  055 

.99  495 

.18  560 

.81  440 

.00  505 

.81  945 

17 

44 

.18  137 

.99  494 

.18  644 

.81  356 

.00  506 

.81  863 

16 

45 

9.18  220 

9.99  492 

9.18  728 

0.81  272 

0.00  508 

0.81  780 

15 

46 

.18  302 

.99  490 

.18  812 

.81  188 

.00  510 

.81  698 

14 

47 

.18  383 

.99  488 

.18  896 

.81  104 

.00  512 

.81  617 

13 

48 

.18  465 

.99  486 

.18  979 

.81  021 

.00  514 

.81  535 

12 

49 

.18  547 

.99  484 

.19  063 

.80  937 

.00  516 

.81  453 

11 

50 

9.18  628 

9.99  482 

9.19  146 

0.80  854 

0.00  518 

0.81  372 

10 

51 

.18  709 

.99  480 

.19  229 

.80  771 

.00  520 

.81  291 

9 

52 

.18  790 

.99  478 

.19  312 

.80  688 

.00  522 

.81  210 

8 

53 

.18  871 

.99  476 

.19  395 

.80  605 

.00  524 

.81  129 

7 

54 

.18  952 

.99  474 

.19  478 

.80  522 

.00  526 

.81  048 

6 

55 

9.19  033 

9.99  472 

9.19  561 

0.80  439 

0.00  528 

0.80  967 

5 

56 

.19  113 

.99  470 

.19  643 

.80  357 

.00  530 

.80  887 

4 

57 

.19  193 

.99  468 

.19  725 

.80  275 

.00  532 

.80  807 

3 

58 

.19  273 

.99  466 

.19  807 

.80  193 

.00  534 

.80  727 

2 

59 

.19  353 

.99  464 

.19  889 

.80  111 

.00  536 

.80  647 

1 

60 

9.19  433 

9.99  462 

9.19  971 

0.80  029 

0.00  538 

0.80  567 
Sec 

0 

Cos 

Sin 

Cot 

Tan 

Csc 

/ 

98°  (278°) 


(261°)  81° 


Table  4.    Trigonometric  Logarithms 


205 


9°  (189°) 

(350°)  170° 

' 

Sin 

Cos 

Tan 

Cot 

Sec 

Csc 

0 

9.19  433 

9.99  462 

9.19  971 

0.80  029 

0.00  538 

0.80  567 

60 

1 

.19  513 

.99  460 

.20  053 

.79  947 

.00  540 

.80  487 

59 

2 

.19  592 

.99  458 

.20  134 

.79  866 

.00  542 

.80  408 

58 

3 

.19  672 

.99  456 

.20  216 

.79  784 

.00  544 

.80  328 

57 

4 

.19  751 

.99  454 

.20  297 

.79  703 

.00  546 

.80  249 

56 

5 

9.19  830 

9.99  452 

9.20  378 

0.79  622 

0.00  548 

0.80  170 

55 

6 

.19  909 

.99  450 

.20  459 

.79  541 

.00  550 

.80  091 

54 

7 

.19  988 

.99  448 

.20  540 

.79  460 

.00  552 

.80  012 

53 

8 

.20  067 

.99  446 

.20  621 

.79  379 

.00  554 

.79  933 

52 

9 

.20  145 

.99  444 

.20  701 

.79  299 

.00  556 

.79  855 

51 

10 

9.20  223 

9.99  442 

9.20  782 

0.79  218 

0.00  558 

0.79  777 

50 

11 

.20  302 

.99  440 

.20  862 

.79  138 

.00  560 

.79  698 

49 

12 

.20  380 

.99  438 

.20  942 

.79  058 

.00  562 

.79  620 

48 

13 

.20  458 

.99  436 

.21  022 

.78  978 

.00  564 

.79  542 

47 

14 

.20  535 

.99  434 

.21  102 

.78  898 

.00  566 

.79  465 

46 

15 

9.20  613 

9.99  432 

9.21  182 

0.78  818 

0.00  568 

0.79  387 

45 

16 

.20  691 

.99  429 

.21  261 

.78  739 

.00  571 

.79  309 

44 

17 

.20  768 

.99  427 

.21  341 

.78  659 

.00  573 

.79  232 

43 

18 

.20  845 

.99  425 

.21  420 

.78  580 

.00  575 

.79  155 

42 

19 

.20  922 

.99  423 

.21  499 

.78  501 

.00  577 

.79  078 

41 

20 

9.20  999 

9.99  421 

9.21  578 

0.78  422 

0.00  579 

0.79  001 

40 

21 

.21  076 

.99  419 

.21  657 

.78  343 

.00  581 

.78  924 

39 

22 

.21  153 

.99  417 

.21  736 

.78  264 

.00  583 

.78  847 

38 

23 

.21  229 

.99  415 

.21  814 

.78  186 

.00  585 

.78  771 

37 

24 

.21  306 

.99  413 

.21  893 

.78  107 

.00  587 

.78  694 

36 

25 

9.21  382 

9.99  411 

9.21  971 

0.78  029 

0.00  589 

0.78  618 

35 

26 

.21  458 

.99  409 

.22  049 

.77  951 

.00  591 

.78  542 

34 

27 

.21  534 

.99  407 

.22  127 

.77  873 

.00  593 

.78  466 

33 

28 

.21  610 

.99  404 

.22  205 

.77  795 

.00  596 

.78  390 

32 

29 

.21  685 

.99  402 

.22  283 

.77  717 

.00  598 

.78  315 

31 

30 

9.21  761 

9.99  400 

9.22  361 

0.77  639 

0.00  600 

0.78  239 

30 

31 

.21  836 

.99  398 

.22  438 

.77  562 

.00  602 

.78  164 

29 

32 

.21  912 

.99  396 

.22  516 

.77  484 

.00  604 

.78  088 

28 

33 

.21  987 

.99  394 

.22  593 

.77  407 

.00  606 

.78  013 

27 

34 

.22  062 

.99  392 

.22  670 

.77  330 

.00  608 

.77  938 

26 

35 

9.22  137 

9.99  390 

9.22  747 

0.77  253 

0.00  610 

0.77  863 

25 

36 

.22  211 

.99  388 

.22  824 

.77  176 

.00  612 

.77  789 

24 

37 

.22  286 

.99  385 

.22  901 

.77  099 

.00  615 

.77  714 

23 

38 

.22  361 

.99  383 

.22  977 

.77  023 

.00  617 

.77  639 

22 

39 

.22  435 

.99  381 

.23  054 

.76  946 

.00  619 

.77  565 

21 

40 

9.22  509 

9.99  379 

9.23  130 

0.76  870 

0.00  621 

0.77  491 

20 

41 

.22  583 

.99  377 

.23  206 

.76  794 

.00  623 

.77  417 

19 

42 

.22  657 

.99  375 

.23  283 

.76  717 

.00  625 

.77  343 

18 

43 

.22  731 

.99  372 

.23  359 

.76  641 

.00  628 

.77  269 

17 

44 

.22  805 

.99  370 

.23  435 

.76  565 

.00  630 

.77  195 

16 

45 

9.22  878 

9.99  368 

9.23  510 

0.76  490 

0.00  632 

0.77  122 

15 

46 

.22  952 

.99  366 

.23  586 

.76  414 

.00  634 

.77  048 

14 

47 

.23  025 

.99  364 

.23  661 

.76  339 

.00  636 

.76  975 

13 

48 

.23  098 

.99  362 

.23  737 

.76  263 

.00  638 

.76  902 

12 

49 

.23  171 

.99  359 

.23  812 

.76  188 

.00  641 

.76  829 

11 

50 

9.23  244 

9.99  357 

9.23  887 

0.76  113 

0.00  643 

0.76  756 

10 

51 

.23  317 

.99  355 

.23  962 

.76  038 

.00  645 

.76  683 

9 

52 

.23  390 

.99  353 

.24  037 

.75  963 

.00  647 

.76  610 

8 

53 

.23  462 

.99  351 

.24  112 

.75  888 

.00  649 

.76  538 

7 

54 

.23  535 

.99  348 

.24  186 

.75  814 

.00  652 

.76  465 

6 

55 

9.23  607 

9.99  346 

9.24  261 

0.75  739 

0.00  654 

0.76  393 

5 

56 

.23  679 

.99  344 

.24  335 

.75  665 

.00  656 

.76  321 

4 

57 

.23  752 

.99  342 

.24  410 

.75  590 

.00  658 

.76  248 

3 

58 

.23  823 

.99  340 

.24  484 

.75  516 

.00  660 

.76  177 

2 

59 

.23  895 

.99  337 

.24  558 

.75  442 

.00  663 

.76  105 

1 

60 

9.23  967 

9.99  335 

9.24  632 

0.75  368 

0.00  665 

0.76  033 

0 

Cos 

Sin 

Cot 

Tan 

Csc 

Sec 

99°  (279°) 


(260°)  80^ 


206 


Table  4.    Trigonometric  Logarithms 


10°  (190°) 

(349°)  169° 

/ 

Sin 

Cos 

Tan 

Cot 

Sec 

Csc 

0 

9.23  967 

9.99  335 

9.24  632 

0.75  368 

0.00  665 

0.76  033 

60 

1 

.24  039 

.99  333 

.24  706 

.75  294 

.00  667 

.75  961 

59 

2 

.24  110 

.99  331 

.24  779 

.75  221 

.00  669 

.75  890 

58 

3 

.24  181 

.99  328 

.24  853 

.75  147 

.00  672 

.75  819 

57 

4 

.24  253 

.99  326 

.24  926 

.75  074 

.00  674 

.75  747 

56 

5 

9.24  324 

9.99  324 

9.25  000 

0.75  000 

0.00  676 

0.75  676 

55 

6 

.24  395 

.99  322 

.25  073 

.74  927 

.00  678 

.75  605 

54 

7 

.24  466 

.99  319 

.25  146 

.74  854 

.00  681 

.75  534 

53 

8 

.24  536 

.99  317 

.25  219 

.74  781 

.00  683 

.75  464 

52 

9 

.24  607 

.99  315 

.25  292 

.74  708 

.00  685 

.75  393 

51 

10 

9.24  677 

9.99  313 

9.25  365 

0.74  635 

0.00  687 

0.75  323 

50 

11 

.24  748 

.99  310 

.25  437 

.74  563 

.00  690 

.75  252 

49 

12 

.24  818 

.99  308 

.25  510 

.74  490 

.00  692 

.75  182 

48 

13 

.24  888 

.99  306 

.25  582 

.74  418 

.00  694 

.75  112 

47 

14 

.24  958 

.99  304 

.25  655 

.74  345 

.00  696 

.75  042 

46 

15 

9.25  028 

9.99  301 

9.25  727 

0.74  273 

0.00  699 

0.74  972 

45 

16 

.25  098 

.99  299 

.25  799 

.74  201 

.00  701 

.74  902 

44 

17 

.25  168 

.99  297 

.25  871 

.74  129 

.00  703 

.74  832 

43 

18 

.25  237 

.99  294 

.25  943 

.74  057 

.00  706 

.74  763 

42 

19 

.25  307 

.99  292 

.26  015 

.73  985 

.00  708 

.74  693 

41 

20 

9.25  376 

9.99  290 

9.26  086 

0.73  914 

0.00  710 

0.74  624 

40 

21 

.25  445 

.99  288 

.26  158 

.73  842 

.00  712 

.74  555 

39 

22 

.25  514 

.99  285 

.26  229 

.73  771 

.00  715 

.74  486 

38 

23 

.25  583 

.99  283 

.26  301 

.73  699 

.00  717 

.74  417 

37 

24 

.25  652 

.99  281 

.26  372 

.73  628 

.00  719 

.74  348 

36 

25 

9.25  721 

9.99  278 

9.26  443 

0.73  557 

0.00  722 

0.74  279 

35 

26 

.25  790 

.99  276 

.26  514 

.73  486 

.00  724 

.74  210 

34 

27 

.25  858 

.99  274 

.26  585 

.73  415 

.00  726 

.74  142 

33 

28 

.25  927 

.99  271 

.26  655 

.73  345 

.00  729 

.74  073 

32 

29 

.25  995 

.99  269 

.26  726 

.73  274 

.00  731 

.74  005 

31 

30 

9.26  063 

9.99  267 

9.26  797 

0.73  203 

0.00  733 

0.73  937 

30 

31 

.26  131 

.99  264 

.26  867 

.73  133 

.00  736 

.73  869 

29 

32 

.26  199 

.99  262 

.26  937 

.73  063 

.00  738 

.73  801 

28 

33 

.26  267 

.99  260 

.27  008 

.72  992 

.00  740 

.73  733 

27 

34 

.26  335 

.99  257 

.27  078 

.72  922 

.00  743 

.73  665 

26 

35 

9.26  403 

9.99  255 

9.27  148 

0.72  852 

0.00  745 

0.73  597 

25 

36 

.26  470 

.99  252 

.27  218 

.72  782 

.00  748 

.73  530 

24 

37 

.26  538 

.99  250 

.27  288 

.72  712 

.00  750 

.73  462 

23 

38 

.26  605 

.99  248 

.27  357 

.72  643 

.00  752 

.73  395 

22 

39 

.26  672 

.99  245 

.27  427 

.72  573 

.00  755 

.73  328 

21 

40 

9.26  739 

9.99  243 

9.27  496 

0.72  504 

0.00  757 

0.73  261 

20 

41 

.26  806 

.99  241 

.27  566 

.72  434 

.00  759 

.73  194 

19 

42 

.26  873 

.99  238 

.27  635 

.72  365 

.00  762 

.73  127 

18 

43 

.26  940 

.99  236 

.27  704 

.72  296 

.00  764 

.73  060 

17 

44 

.27  007 

.99  233 

.27  773 

.72  227 

.00  767 

.72  993 

16 

45 

9.27  073 

9.99  231 

9.27  842 

0.72  158 

0.00  769 

0.72  927 

15 

46 

.27  140 

.99  229 

.27  911 

.72  089 

.00  771 

.72  860 

14 

47 

.27  206 

.99  226 

.27  980 

.72  020 

.00  774 

.72  794 

13 

48 

.27  273 

.99  224 

.28  049 

.71  951 

.00  776 

.72  727 

12 

49 

.27  339 

.99  221 

.28  117 

.71  883 

.00  779 

.72  661 

11 

50 

9.27  405 

9.99  219 

9.28  186 

0.71  814 

0.00  781 

0.72  595 

10 

51 

.27  471 

.99  217 

.28  254 

.71  746 

.00  783 

.72  529 

9 

52 

.27  537 

.99  214 

.28  323 

.71  677 

.00  786 

.72  463 

8 

53 

.27  602 

.99  212 

.28  391 

.71  609 

.00  788 

.72  398 

7 

54 

.27  668 

.99  209 

.28  459 

.71  541 

.00  791 

.72  332 

6 

55 

9.27  734 

9.99  207 

9.28  527 

0.71  473 

0.00  793 

0.72  266 

5 

56 

.27  799 

.99  204 

.28  595 

.71  405 

.00  796 

.72  201 

4 

57 

.27  864 

.99  202 

.28  662 

.71  338 

.00  798 

.72  136 

3 

58 

.27  930 

.99  200 

.28  730 

.71  270 

.00  800 

.72  070 

2 

59 

.27  995 

.99  197 

.28  798 

.71  202 

.00  803 

.72  005 

1 

60 

9.28  060 

9.99  195 

9.28  865 

0.71  135 

0.00  805 

0.71  940 

0 

Cos 

Sin 

Cot 

Tan 

Csc 

Sec      '   1 

100°  (280°) 


(259°)  79= 


Table  4.    Tiigonometric  Logarithms 


207 


11°  (191°) 

(348° 

)  168° 

' 

Sin 

Cos 

Tan 

Cot 

Sec 

Csc 

0 

9.28  060 

9.99  195 

9.28  865 

0.71  135 

0.00  805 

0.71  940 

60 

1 

.28  125 

.99  192 

.28  933 

.71  067 

.00  808 

.71  875 

59 

2 

.28  190 

.99  190 

.29  000 

.71  000 

.00  810 

.71  810 

58 

3 

.28  254 

.99  187 

.29  067 

.70  933 

.00  813 

.71  746 

57 

4 

.28  319 

.99  185 

.29  134 

.70  866 

.00  815 

.71  681 

56 

5 

9.28  384 

9.99  182 

9.29  201 

0.70  799 

0.00  818 

0.71  616 

55 

6 

.28  448 

.99  180 

.29  268 

.70  732 

.00  820 

.71  552 

54 

7 

.28  512 

.99  177 

.29  335 

.70  665 

.00  823 

.71  488 

53 

8 

.28  577 

.99  175 

.29  402 

.70  598 

.00  825 

.71  423 

52 

9 

.28  641 

.99  172 

.29  468 

.70  532 

.00  828 

.71  359 

51 

10 

9.28  705 

9.99  170 

9.29  535 

0.70  465 

0.00  830 

0.71  295 

50 

11 

.28  769 

.99  167 

.29  601 

.70  399 

.00  833 

.71  231 

49 

12 

.28  833 

.99  165 

.29  668 

.70  332 

.00  835 

.71  167 

48 

13 

.28  896 

.99  162 

.29  734 

.70  266 

.00  838 

.71  104 

47 

14 

.28  960 

.99  160 

.29  800 

.70  200 

.00  840 

.71  040 

46 

15 

9.29  024 

9.99  157 

9.29  866 

0.70  134 

0.00  843 

0.70  976 

45 

16 

.29  087 

.99  155 

.29  932 

.70  068 

.00  845 

.70  913 

44 

17 

.29  150 

.99  152 

.29  998 

.70  002 

.00  848 

.70  850 

43 

18 

.29  214 

.99  150 

.30  064 

.69  936 

.00  850 

.70  786 

42 

19 

.29  277 

.99  147 

.30  130 

.69  870 

.00  853 

.70  723 

41 

20 

9.29  340 

9.99  145 

9.30  195 

0.69  805 

0.00  855 

0.70  660 

40 

21 

.29  403 

.99  142 

.30  261 

.69  739 

.00  858 

.70  597 

39 

22 

.29  466 

.99  140 

.30  326 

.69  674 

.00  860 

.70  534 

38 

23 

.29  529 

.99  137 

.30  391 

.69  609 

.00  863 

.70  471 

37 

24 

.29  591 

.99  135 

.30  457 

.69  543 

.00  865 

.70  409 

36 

25 

9.29  654 

9.99  132 

9.30  522 

0.69  478 

0.00  868 

0.70  346 

35 

26 

.29  716 

.99  130 

.30  587 

.69  413 

.00  870 

.70  284 

34 

27 

.29  779 

.99  127 

.30  652 

.69  348 

.00  873 

.70  221 

33 

28 

.29  841 

.99  124 

.30  717 

.69  283 

.00  876 

.70  159 

32 

29 

.29  903 

.99  122 

.30  782 

.69  218 

.00  878 

.70  097 

31 

30 

9.29  966 

9.99  119 

9.30  846 

0.69  154 

0.00  881 

0.70  034 

30 

31 

.30  028 

.99  117 

.30  911 

.69  089 

.00  883 

.69  972 

29 

32 

.30  090 

.99  114 

.30  975 

.69  025 

.00  886 

.69  910 

28 

33 

.30  151 

.99  112 

.31  040 

.68  960 

.00  888 

.69  849 

27 

34 

.30  213 

.99  109 

.31  104 

.68  896 

.00  891 

.69  787 

26 

35 

9.30  275 

9.99  106 

9.31  168 

0.68  832 

0.00  894 

0.69  725 

25 

36 

.30  336 

.99  104 

.31  233 

.68  767 

.00  896 

.69  664 

24 

37 

.30  398 

.99  101 

.31  297 

.68  703 

.00  899 

.69  602 

23 

38 

.30  459 

.99  099 

.31  361 

.68  639 

.00  901 

.69  541 

22 

39 

.30  521 

.99  096 

.31  425 

.68  575 

.00  904 

.69  479 

21 

40 

9.30  582 

9.99  093 

9.31  489 

0.68  511 

0.00  907 

0.69  418 

20 

41 

.30  643 

.99  091 

.31  552 

.68  448 

.00  909 

.69  357 

19 

42 

.30  704 

.99  088 

.31  616 

.68  384 

.00  912 

.69  296 

18 

43 

.30  765 

.99  086 

.31  679 

.68  321 

.00  914 

.69  235 

17 

44 

.30  826 

.99  083 

.31  743 

.68  257 

.00  917 

.69  174 

16 

45 

9.30  887 

9.99  080 

9.31  806 

0.68  194 

0.00  920 

0.69  113 

15 

46 

.30  947 

.99  078 

.31  870 

.68  130 

.00  922 

.69  053 

14 

47 

.31  008 

.99  075 

.31  933 

.68  067 

.00  925 

.68  992 

13 

48 

.31  068 

.99  072 

.31  996 

.68  004 

.00  928 

.68  932 

12 

49 

.31  129 

.99  070 

.32  059 

.67  941 

.00  930 

.68  871 

11 

50 

9.31  189 

9.99  067 

9.32  122 

0.67  878 

0.00  933 

0.68  811 

10 

51 

.31  250 

.99  064 

.32  185 

.67  815 

.00  936 

.68  750 

9 

52 

.31  310 

.99  062 

.32  248 

.67  752 

.00  938 

.68  690 

8 

53 

.31  370 

.99  059 

.32  311 

.67  689 

.00  941 

.68  630 

7 

54 

.31  430 

.99  056 

.32  373 

.67  627 

.00  944 

.68  570 

6 

55 

9.31  490 

9.99  054 

9.32  436 

0.67  564 

0.00  946 

0.68  510 

5 

56 

.31  549 

.99  051 

.32  498 

.67  502 

.00  949 

.68  451 

4 

57 

.31  609 

.99  048 

.32  561 

.67  439 

.00  952 

.68  391 

3 

58 

.31  669 

.99  046 

.32  623 

.67  377 

.00  954 

.68  331 

2 

59 

.31  728 

.99  043 

.32  685 

.67  315 

.00  957 

.68  272 

1 

60 

9.31  788 

9.99  040 

9.32  747 

0.67  253 

0.00  960 

0.68  212 

0 

Cos 

Sin 

Cot 

Tan 

Csc 

Sec 

/ 

101°  (281°) 


(258°)  78' 


208 


Table  4.    Trigonometric  Logarithms 


12°  (192°) 

(347°) 

167° 

' 

Sin 

Cos 

Tan 

Cot 

Sec 

Csc 

0 

9.31  788 

9.99  040 

9.32  747 

0.67  253 

0.00  960 

0.68  212 

60 

1 

.31  847 

.99  038 

.32  810 

.67  190 

.00  962 

.68  153 

59 

2 

.31  907 

.99  035 

.32  872 

.67  128 

.00  965 

.68  093 

58 

3 

.31  966 

.99  032 

.32  933 

.67  067 

.00  968 

.68  034 

57 

4 

.32  025 

.99  030 

.32  995 

.67  005 

.00  970 

.67  975 

56 

5 

9.32  084 

9.99  027 

9.33  057 

0.66  943 

0.00  973 

0.67  916 

55 

6 

.32  143 

.99  024 

.33  119 

.66  881 

.00  976 

.67  857 

54 

7 

.32  202 

.99  022 

.33  180 

.66  820 

-.00  978 

.67  798 

53 

8 

.32  261 

.99  019 

.33  242 

.66  758 

.00  981 

.67  739 

52 

9 

.32  319 

.99  016 

.33  303 

.66  697 

.00  984 

.67  681 

51 

10 

9.32  378 

9.99  013 

9.33  365 

0.66  635 

0.00  987 

0.67  622 

50 

11 

.32  437 

.99  011 

.33  426 

.66  574 

.00  989 

.67  563 

49 

12 

.32  495 

.99  008 

.33  487 

.66  513 

.00  992 

.67  505 

48 

13 

.32  553 

.99  005 

.33  548 

.66  452 

.00  995 

.67  447 

47 

14 

.32  612 

.99  002 

.33  609 

.66  391 

.00  998 

.67  388 

46 

15 

9.32  670 

9.99  000 

9.33  670 

0.66  330 

0.01  000 

0.67  330 

45 

16 

.32  728 

.98  997 

.33  731 

.66  269 

.01  003 

.67  272 

44 

17 

.32  786 

.98  994 

.33  792 

.66  208 

.01  006 

.67  214 

43 

18 

.32  844 

.98  991 

.33  853 

.66  147 

.01  009 

.67  156 

42 

19 

.32  902 

.98  989 

.33  913 

.66  087 

.01  Oil 

.67  098 

41 

20 

9.32  960 

9.98  986 

9.33  974 

0.66  026 

0.01  014 

0.67  040 

40 

21 

.33  018 

.98  983 

.34  034 

.65  966 

.01  017 

.66  982 

39 

22 

.33  075 

.98  980 

.34  095 

.65  905 

.01  020 

.66  925 

38 

23 

.33  133 

.98  978 

.34  155 

.65  845 

.01  022 

.66  867 

37 

24 

.33  190 

.98  975 

.34  215 

.65  785 

.01  025 

.66  810 

36 

25 

9.33  248 

9.98  972 

9.34  276 

0.65  724 

0.01  028 

0.66  752 

35 

26 

.33  305 

.98  969 

.34  336 

.65  664 

.01  031 

.66  695 

34 

27 

.33  362 

.98  967 

.34  396 

.65  604 

.01  033 

.66  638 

33 

28 

.33  420 

.98  964 

.34  456 

.65  544 

.01  036 

.66  580 

32 

29 

.33  477 

.98  961 

.34  516 

.65  484 

.01  039 

.66  523 

31 

30 

9.33  534 

9.98  958 

9.34  576 

0.65  424 

0.01  042 

0.66  466 

30 

31 

.33  591 

.98  955 

.34  635 

.65  365 

.01  045 

.66  409 

29 

32 

.33  647 

.98  953 

.34  695 

.65  305 

.01  047 

.66  353 

28 

33 

.33  704 

.98  950 

.34  755 

.65  245 

.01  050 

.66  296 

27 

34 

.33  761 

.98  947 

.34  814 

.65  186 

.01  053 

.66  239 

26 

35 

9.33  818 

9.98  944 

9.34  874 

0.65  126 

0.01  056 

0.66  182 

25 

36 

.33  874 

.98  941 

.34  933 

.65  067 

.01  059 

.66  126 

24 

37 

.33  931 

.98  938 

.34  992 

.65  008 

.01  062 

.66  069 

23 

38 

.33  987 

.98  936 

.35  051 

.64  949 

.01  064 

.66  013 

22 

39 

.34  043 

.98  933 

.35  111 

.64  889 

.01  067 

.65  957 

21 

40 

9.34  100 

9.98  930 

9.35  170 

0.64  830 

0.01  070 

0.65  900 

20 

41 

.34  156 

.98  927 

.35  229 

.64  771 

.01  073 

.65  844 

19 

42 

.34  212 

.98  924 

.35  288 

.64  712 

.01  076 

.65  788 

18 

43 

.34  268 

.98  921 

.35  347 

.64  653 

.01  079 

.65  732 

17 

44 

.34  324 

.98  919 

.35  405 

.64  595 

.01  081 

.65  676 

16 

45 

9.34  380 

9.98  916 

9.35  464 

0.64  536 

0.01  084 

0.65  620  . 

15 

46 

.34  436 

.98  913 

.35  523 

.64  477 

.01  087 

.65  564 

14 

47 

.34  491 

.98  910 

.35  581 

.64  419 

.01  090 

.65  509 

13 

48 

.34  547 

.98  907 

.35  640 

.64  360 

.01  093 

.65  453 

12 

49 

.34  602 

.98  904 

.35  698 

.64  302 

.01  096 

.65  398 

11 

50 

9.34  658 

9.98  901 

9.35  757 

0.64  243 

0.01  099 

0.65  342 

10 

51 

.34  713 

.98  898 

.35  815 

.64  185 

.01  102 

.65  287 

9 

52 

.34  769 

.98  896 

.35  873 

.64  127 

.01  104 

.65  231 

8 

53 

.34  824 

.98  893 

.35  931 

.64  069 

.01  107 

.65  176 

7 

54 

.34  879 

.98  890 

.35  989 

.64  011 

.01  110 

.65  121 

6 

55 

9.34  934 

9.98  887 

9.36  047 

0.63  953 

0.01  113 

0.65  066 

5 

56 

.34  989 

.98  884 

.36  105 

.63  895 

.01  116 

.65  011 

4 

57 

.35  044 

.98  881 

.36  163 

.63  837 

.01  119 

.64  956 

3 

58 

.35  099 

.98  878 

.36  221 

.63  779 

.01  122 

.64  901 

2 

59 

.35  154 

.98  875 

.36  279 

.63  721 

.01  125 

.64  846 

1 

60 

9.35  209 

9.98  872 

9.36  336 

0.63  664 

0.01  128 

0.64  791 

0 

Cos 

Sin 

Cot 

Tan 

Csc 

Sec 

/ 

102°  (282°) 


(257°)  77' 


Table  4.    Trigonometric  Logarithms 


209 


13°  (193°) 

(346°) 

166° 

/ 

Sin 

Cos 

Tan 

Cot 

Sec 

Csc 

0 

9.35  209 

9.98  872 

9.36  336 

0.63  664 

0.01  128 

0.64  791 

60 

1 

.35  263 

.98  869 

.36  394 

.63  606 

.01  131 

.64  737 

59 

2 

.35  318 

.98  867 

.36  452 

.63  548 

.01  133 

.64  682 

58 

3 

.35  373 

.98  864 

.36  509 

.63  491 

.01  136 

.64  627 

57 

4 

.35  427 

.98  861 

.36  566 

.63  434 

.01  139 

.64  573 

56 

5 

9.35  481 

9.98  858 

9.36  624 

0.63  376 

0.01  142 

0.64  519 

55 

6 

.35  536 

.98  855 

.36  681 

.63  319 

.01  145 

.64  464 

54 

7 

.35  590 

.98  852 

.36  738 

.63  262 

.01  148 

.64  410 

53 

8 

.35  644 

.98  849 

.36  795 

.63  205 

.01  151 

.64  356 

52 

9 

.35  698 

.98  846 

.36  852 

.63  148 

.01  154 

.64  302 

51 

10 

9.35  752 

9.98  843 

9.36  909 

0.63  091 

0.01  157 

0.64  248 

50 

11 

.35  806 

.98  840 

.36  966 

.63  034 

.01  160 

.64  194 

49 

12 

.35  860 

.98  837 

.37  023 

.62  977 

.01  163 

.64  140 

48 

13 

.35  914 

.98  834 

.37  080 

.62  920 

.01  166 

.64  086 

47 

14 

.35  968 

.98  831 

.37  137 

.62  863 

.01  169 

.64  032 

46 

15 

9.36  022 

9.98  828 

9.37  193 

0.62  807 

0.01  172 

0.63  978 

45 

16 

.36  075 

.98  825 

.37  250 

.62  750 

.01  175 

.63  925 

44 

17 

.36  129 

.98  822 

.37  306 

.62  694 

.01  178 

.63  871 

43 

18 

.36  182 

.98  819 

.37  363 

.62  637 

.01  181 

.63  818 

42 

19 

.36  236 

.98  816 

.37  419 

.62  581 

.01  184 

.63  764 

41 

20 

9.36  289 

9.98  813 

9.37  476 

0.62  524 

0.01  187 

0.63  711 

40 

21 

.36  342 

.98  810 

.37  532 

.62  468 

.01  190 

.63  658 

39 

22 

.36  395 

.98  807 

.37  588 

.62  412 

.01  193 

.63  605 

38 

23 

.36  449 

.98  804 

.37  644 

.62  356 

.01  196 

.63  551 

37 

24 

.36  502 

.98  801 

.37  700 

.62  300 

.01  199 

.63  498 

36 

25 

9.36  555 

9.98  798 

9.37  756 

0.62  244 

0.01  202 

0.63  445 

35 

26 

.36  608 

.98  795 

.37  812 

.62  188 

.01  205 

.63  392 

34 

27 

.36  660 

.98  792 

.37  868 

.62  132 

.01  208 

.63  340 

33 

28 

.36  713 

.98  789 

.37  924 

.62  076 

.01  211 

.63  287 

32 

29 

.36  766 

.98  786 

.37  980 

.62  020 

.01  214 

.63  234 

31 

30 

9.36  819 

9.98  783 

9.38  035 

0.61  965 

0.01  217 

0.63  181 

30 

31 

.36  871 

.98  780 

.38  091 

.61  909 

.01  220 

.63  129 

29 

32 

.36  924 

.98  777 

.38  147 

.61  853 

.01  223 

.63  076 

28 

33 

.36  976 

.98  774 

.38  202 

.61  798 

.01  226 

.63  024 

27 

34 

.37  028 

.98  771 

.38  257 

.61  743 

.01  229 

.62  972 

26 

35 

9.37  081 

9.98  768 

9.38  313 

0.61  687 

0.01  232 

0.62  919 

25 

36 

.37  133 

.98  765 

.38  368 

.61  632 

.01  235 

.62  867 

24 

37 

.37  185 

.98  762 

.38  423 

.61  577 

.01  238 

.62  815 

23 

38 

.37  237 

.98  759 

.38  479 

.61  521 

.01  241 

.62  763 

22 

39 

.37  289 

.98  756 

.38  534 

.61  466 

.01  244 

.62  711 

21 

40 

9.37  341 

9.98  753 

9.38  589 

0.61  411 

0.01  247 

0.62  659 

20 

41 

.37  393 

.98  750 

.38  644 

.61  356 

.01  250 

.62  607 

19 

42 

.37  445 

.98  746 

.38  699 

.61  301 

.01  254 

.62  555 

18 

43 

.37  497 

.98  743 

.38  754 

.61  246 

.01  257 

.62  503 

17 

44 

.37  549 

.98  740 

.38  808 

.61  192 

.01  260 

.62  451 

16 

45 

9.37  600 

9.98  737 

9.38  863 

0.61  137 

0.01  263 

0.62  400 

15 

46 

.37  652 

.98  734 

.38  918 

.61  082 

.01  266 

.62  348 

14 

47 

.37  703 

.98  731 

.38  972 

.61  028 

.01  269 

.62  297 

13 

48 

.37  755 

.98  728 

.39  027 

.60  973 

.01  272 

.62  245 

12 

49 

.37  806 

.98  725 

.39  082 

.60  918 

.01  275 

.62  194 

11 

50 

9.37  858 

9.98  722 

9.39  136 

0.60  864 

0.01  278 

0.62  142 

10 

51 

.37  909 

.98  719 

.39  190 

.60  810 

.01  281 

.62  091 

9 

52 

.37  960 

.98  715 

.39  245 

.60  755 

.01  285 

.62  040 

8 

53 

.38  011 

.98  712 

.39  299 

.60  701 

.01  288 

.61  989 

7 

54 

.38  062 

.98  709 

.39  353 

.60  647 

.01  291 

.61  938 

6 

55 

9.38  113 

9.98  706 

9.39  407 

0.60  593 

0.01  294 

0.61  887 

5 

56 

.38  164 

.98  703 

.39  461 

.60  539 

.01  297 

.61  836 

4 

57 

.38  215 

.98  700 

.39  515 

.60  485 

.01  300 

.61  785 

3 

58 

.38  266 

.98  697 

.39  569 

.60  431 

.01  303 

.61  734 

2 

59 

.38  317 

.98  694 

.39  623 

.60  377 

.01  306 

.61  683 

1 

60 

9.38  368 

9.98  690 

9.39  677 

0.60  323 

0.01  310 

0.61  632 

0 

Cos   ! 

Sin 

Cot 

Tan 

Csc 

Sec 

/ 

103°  (283°) 


(256°)  76° 


210 


Table  4.    Trigonometric  Logarithms 


14°  (194°) 

(345°)  165° 

' 

Sin 

Cos 

Tan 

Cot 

Sec 

Csc 

0 

9.38  368 

9.98  690 

9.39  677 

0.60  323 

0.01  310 

0.61  632 

60 

1 

.38  418 

.98  687 

.39  731 

.60  269 

.01  313 

.61  582 

59 

2 

.38  469 

.98  684 

.39  785 

.60  215 

.01  316 

.61  531 

58 

3 

.38  519 

.98  681 

.39  838 

.60  162 

.01  319 

.61  481 

57 

4 

.38  570 

.98  678 

.39  892 

.60  108 

.01  322 

.61  430 

56 

5 

9.38  620 

9.98  675 

9.39  945 

0.60  055 

0.01  325 

0.61  380 

55 

6 

.38  670 

.98  671 

.39  999 

.60  001 

.01  329 

..61  330 

54 

7 

.38  721 

.98  668 

.40  052 

.59  948 

.01  332 

.61  279 

53 

8 

.38  771 

.98  665 

.40  106 

.59  894 

.01  335 

.61  229 

52 

9 

.38  821 

.98  662 

.40  159 

.59  841 

.01  338 

.61  179 

51 

10 

9.38  871 

9.98  659 

9.40  212 

0.59  788 

0.01  341 

0.61  129 

50 

11 

.38  921 

.98  656 

.40  266 

.59  734 

.01  344 

.61  079 

49 

12 

.38  971 

.98  652 

.40  319 

.59  681 

.01  348 

.61  029 

48 

13 

.39  021 

.98  649 

.40  372 

.59  628 

.01  351 

.60  979 

47 

14 

.39  071 

.98  646 

.40  425 

.59  575 

.01  354 

.60  929 

46 

15 

9.39  121 

9.98  643 

9.40  478 

0.59  522 

0.01  357 

0.60  879 

45 

16 

.39  170 

.98  640 

.40  531 

.59  469 

.01  360 

.60  830 

44 

17 

.39  220 

.98  636 

.40  584 

:59  416 

.01  364 

.60  780 

43 

18 

.39  270 

.98  633 

.40  636 

.59  364 

.01  367 

.60  730 

42 

19 

.39  319 

.98  630 

.40  689 

.59  311 

.01  370 

.60  681 

41 

20 

9.39  369 

9.98  627 

9.40  742 

0.59  258 

0.01  373 

0.60  631 

40 

21 

.39  418 

.98  623 

.40  795 

.59  205 

.01  377 

.60  582 

39 

22 

.39  467 

.98  620 

.40  847 

.59  153 

.01  380 

.60  533 

38 

23 

.39  517 

.98  617 

.40  900 

.59  100 

.01  383 

.60  483 

37 

24 

.39  566 

.98  614 

.40  952 

.59  048 

.01  386 

.60  434 

36 

25 

9.39  615 

9.98  610 

9.41  005 

0.58  995 

0.01  390 

0.60  385 

35 

26 

.39  664 

.98  607 

.41  057 

.58  943 

.01  393 

.60  336 

34 

27 

.39  713 

.98  604 

.41  109 

.58  891 

.01  396 

.60  287 

33 

28 

.39  762 

.98  601 

.41  161 

.58  839 

.01  399 

.60  238 

32 

29 

.39  811 

.98  597 

.41  214 

.58  786 

.01  403 

.60  189 

31 

30 

9.39  860 

9.98  594 

9.41  266 

0.58  734 

0.01  406 

0.60  140 

30 

31 

.39  909 

.98  591 

.41  318 

.58  682 

.01  409 

.60  091 

29 

32 

.39  958 

.98  588 

.41  370 

.58  630 

.01  412 

.60  042 

28 

33 

.40  006 

.98  584 

.41  422 

.58  578 

.01  416 

.59  994 

27 

34 

.40  055 

.98  581 

.41  474 

.58  526 

.01  419 

.59  945 

26 

35 

9.40  103 

9.98  578 

9.41  526 

0.58  474 

0.01  422 

0.59  897 

25 

36 

.40  152 

.98  574 

.41  578 

.58  422 

.01  426 

.59  848 

24 

37 

.40  200 

.98  571 

.41  629 

.58  371 

.01  429 

.59  800 

23 

38 

.40  249 

.98  568 

.41  681 

.58  319 

.01  432 

.59  751 

22 

39 

.40  297 

.98  565 

.41  733 

.58  267 

.01  435 

.59  703 

21 

40 

9.40  346 

9.98  561 

9.41  784 

0.58  216 

0.01  439 

0.59  654 

20 

41 

.40  394 

.98  558 

.41  836 

.58  164 

.01  442 

.59  606 

19 

42 

.40  442 

.98  555 

.41  887 

.58  113 

.01  445 

.59  558 

18 

43 

.40  490 

.98  551 

.41  939 

.58  061 

.01  449 

.59  510 

17 

44 

.40  538 

.98  548 

.41  990 

.58  010 

.01  452 

.59  462 

16 

45 

9.40  586 

9.98  545 

9.42  041 

0.57  959 

0.01  455 

0.59  414 

15 

46 

.40  634 

.98  541 

:42  093 

.57  907 

.01  459 

.59  366 

14 

47 

.40  682 

.98  538 

.42  144 

.57  856 

.01  462 

.59  318 

13 

48 

.40  730 

.98  535 

.42  195 

.57  805 

.01  465 

.59  270 

12 

49 

.40  778 

.98  531 

.42  246 

.57  754 

.01  469 

.59  222 

11 

50 

9.40  825 

9.98  528 

9.42  297 

0.57  703 

0.01  472 

0.59  175 

10 

51 

.40  873 

.98  525 

.42  348 

.57  652 

.01  475 

.59  127 

9 

52 

.40  921 

.98  521 

.42  399 

.57  601 

.01  479 

.59  079 

8 

53 

.40  968 

.98  518 

.42  450 

.57  550 

.01  482 

.59  032 

7 

54 

.41  016 

.98  515 

.42  501 

.57  499 

.01  485 

.58  984 

6 

55 

9.41  063 

9.98  511 

9.42  552 

0.57  448 

0.01  489 

0.58  937 

5 

56 

.41  111 

.98  508 

.42  603 

.57  397 

.01  492 

.58  889 

4 

57 

.41  158 

.98  505 

.42  653 

.57  347 

.01  495 

.58  842 

3 

58 

.41  205 

.98  501 

.42  704 

.57  296 

.01  499 

.58  795 

2 

59 

.41  252 

.98  498 

.42  755 

.57  245 

.01  502 

.58  748 

1 

60 

9.41  300 

9.98  494 

9.42  805 

0.57  195 

0.01  506 

0.58  700 

0 

Cos 

Sin 

Cot 

Tan 

Csc 

Sec 

/ 

104°  (284°) 


(255°)  75' 


Table  4.    Trigonometric  Logarithms 


211 


15°  (195°) 


(344°)  164= 


/ 

Sin 

1   Cos 

Tan 

Cot 

Sec 

Csc 

0 

9.41  300 

9.98  494 

9.42  805 

0.57  195 

0.01  506 

0.58  700 

60 

1 

.41  347 

.98  491 

.42  856 

.57  144 

.01  509 

.58  653 

59 

2 

.41  394 

.98  488 

.42  906 

.57  094 

.01  512 

.58  606 

58 

3 

.41  441 

.98  484 

.42  957 

.57  043 

.01  516 

.58  559 

57 

4 

.41  488 

.98  481 

.43  007 

.56  993 

.01  519 

.58  512 

56 

5 

9.41  535 

9.98  477 

9.43  057 

0.56  943 

0.01  523 

0.58  465 

55 

6 

.41  582 

.98  474 

.43  108 

.56  892 

.01  526 

.58  418 

54 

7 

.41  628 

.98  471 

.43  158 

.56  842 

.01  529 

.58  372 

53 

8 

.41  675 

.98  467 

.43  208 

.56  792 

.01  533 

.58  325 

52 

9 

.41  722 

.98  464 

.43  258 

.56  742 

.01  536 

.58  278 

51 

10 

9.41  768 

9.98  460 

9.43  308 

0.56  692 

0.01  540 

0.58  232 

50 

11 

.41  815 

.98  457 

.43  358 

.56  642 

.01  543 

.58  185 

49 

12 

.41  861 

.98  453 

.43  408 

.56  592 

.01  547 

.58  139 

48 

13 

.41  908 

.98  450 

.43  458 

.56  542 

.01  550 

.58  092 

47 

14 

.41  954 

.98  447 

.43  508 

.56  492 

.01  553 

.58  046 

46 

15 

9.42  001 

9.98  443 

9.43  558 

0.56  442 

0.01  557 

0.57  999 

45 

16 

.42  047 

.98  440 

.43  607 

.56  393 

.01  560 

.57  953 

44 

17 

.42  093 

.98  436 

.43  657 

.56  343 

.01  564 

.57  907 

43 

18 

.42  140 

.98  433 

.43  707 

.56  293 

.01  567 

.57  860 

42 

19 

.42  186 

.98  429 

.43  756 

.56  244 

.01  571 

.57  814 

41 

20 

9.42  232 

9.98  426 

9.43  806 

0.56  194 

0.01  574 

0.57  768 

40 

21 

.42  278 

.98  422 

.43  855 

.56  145 

.01  578 

.57  722 

39 

22 

.42  324 

.98  419 

.43  905 

.56  095 

.01  581 

.57  676 

38 

23 

.42  370 

.98  415 

.43  954 

.56  046 

.01  585 

.57  630 

37 

24 

.42  416 

.98  412 

.44  004 

.55  996 

.01  588 

.57  584 

36 

25 

9.42  461 

9.98  409 

9.44  053 

0.55  947 

0.01  591 

0.57  539 

35 

26 

.42  507 

.98  405 

.44  102 

.55  898 

.01  595 

.57  493 

34 

27 

.42  553 

.98  402 

.44  151 

.55  849 

.01  598 

.57  447 

33 

28 

.42  599 

.98  398 

.44  201 

.55  799 

.01  602 

.57  401 

32 

29 

.42  644 

.98  395 

.44  250 

.55  750 

.01  605 

.57  356 

31 

30 

9.42  690 

9.98  391 

9.44  299 

0.55  701 

0.01  609 

0.57  310 

30 

31 

.42  735 

.98  388 

.44  348 

.55  652 

.01  612 

.57  265 

29 

32 

.42  781 

.98  384 

.44  397 

.55  603 

.01  616 

.57  219 

28 

33 

.42  826 

.98  381 

.44  446 

.55  554 

.01  619 

.57  174 

27 

34 

.42  872 

.98  377 

.44  495 

.55  505 

.01  623 

.57  128 

26 

35 

9.42  917 

9.98  373 

9.44  544 

0.55  456 

0.01  627 

0.57  083 

25 

36 

.42  962 

.98  370 

.44  592 

.55  408 

.01  630 

.57  038 

24 

37 

.43  008 

.98  366 

.44  641 

.55  359 

.01  634 

.56  992 

23 

38 

.43  053 

.98  363 

.44  690 

.55  310 

.01  637 

.56  947 

22 

39 

.43  098 

.98  359 

.44  738 

.55  262 

.01  641 

.56  902 

21 

40 

9.43  143 

9.98  356 

9.44  787 

0.55  213 

0.01  644 

0.56  857 

20 

41 

.43  188 

.98  352 

.44  836 

.55  164 

.01  648 

.56  812 

19 

42 

.43  233 

.98  349 

.44  884 

.55  116 

.01  651 

.56  767 

18 

43 

.43  278 

.98  345 

.44  933 

.55  067 

.01  655 

.56  722 

17 

44 

.43  323 

.98  342 

.44  981 

.55  019 

.01  658 

.56  677 

16 

45 

9.43  367 

9.98  338 

9.45  029 

0.54  971 

0.01  662 

0.56  633 

15 

46 

.43  412 

.98  334 

.45  078 

.54  922 

.01  666 

.56  588 

14 

47 

.43  457 

.98  331 

.45  126 

.54  874 

.01  669 

.56  543 

13 

48 

.43  502 

.98  327 

.45  174 

.54  826 

.01  673 

.56  498 

12 

49 

.43  546 

.98  324 

.45  222 

.54  778 

.01  676 

.56  454 

11 

50 

9.43  591 

9.98  320 

9.45  271 

0.54  729 

0.01  680 

0.56  409 

10 

51 

.43  635 

.98  317 

.45  319 

.54  681 

.01  683 

.56  365 

9 

52 

.43  680 

.98  313 

.45  367 

.54  633 

.01  687 

.56  320 

8 

53 

.43  724 

.98  309 

.45  415 

.54  585 

.01  691 

.56  276 

7 

54 

.43  769 

.98  306 

.45  463 

.54  537 

.01  694 

.56  231 

6 

55 

9.43  813 

9.98  302 

9.45  511 

0.54  489 

0.01  698 

0.56  187 

5 

56 

.43  857 

.98  299 

.45  559 

.54  441 

.01  701 

.56  143 

4 

57 

.43  901 

.98  295 

.45  606 

.54  394 

.01  705 

.56  099 

3 

58 

.43  946 

.98  291 

.45  654 

.54  346 

.01  709 

.56  054 

2 

59 

.43  990 

.98  288 

.45  702 

.54  298 

.01  712 

.56  010 

1 

60 

9.44  034 

9.98  284 

9.45  750 

0.54  250 

0.01  716 

0.55  966 

0 

Cos 

Sin 

Cot 

Tan 

Csc 

Sec 

' 

105°  (285°) 


(254°)  74° 


212 


Table  4.    Trigonometric  Logarithms 


16°  (196°) 

(343°)  163° 

/ 

Sin 

Cos 

Tan 

Cot 

Sec 

Csc 

0 

9.44  034 

9.98  284 

9.45  750 

0.54  250 

0.01  716 

0.55  966 

60 

1 

.44  078 

.98  281 

.45  797 

.54  203 

.01  719 

.55  922 

59 

2 

.44  122 

.98  277 

.45  845 

.54  155 

.01  723 

.55  878 

58 

3 

.44  166 

.98  273 

.45  892 

.54  108 

.01  727 

.55  834 

57 

4 

.44  210 

.98  270 

.45  940 

.54  060 

.01  730 

.55  790 

56 

5 

9.44  253 

9.98  266 

9.45  987 

0.54  013 

0.01  734 

0.55  747 

55 

6 

.44  297 

.98  262 

.46  035 

.53  965 

.01  738 

.55  703 

54 

7 

.44  341 

.98  259 

.46  082 

.53  918 

.01  741 

.55  659 

53 

8 

.44  385 

.98  255 

.46  130 

.53  870 

.01  745 

.55  615 

52 

9 

.44  428 

.98  251 

.46  177 

.53  823 

.01  749 

.55  572 

51 

10 

9.44  472 

9.98  248 

9.46  224 

0.53  776 

0.01  752 

0.55  528 

50 

11 

.44  516 

.98  244 

.46  271 

.53  729 

.01  756 

.55  484 

49 

12 

.44  559 

.98  240 

.46  319 

.53  681 

.01  760 

.55  441 

48 

13 

.44  602 

.98  237 

.46  366 

.53  634 

.01  763 

.55  398 

47 

14 

.44  646 

.98  233 

.46  413 

.53  587 

.01  767 

.55  354 

46 

15 

9.44  689 

9.98  229 

9.46  460 

0.53  540 

0.01  771 

0.55  311 

45 

16 

.44  733 

.98  226 

.46  507 

.53  493 

.01  774 

.55  267 

44 

17 

.44  776 

.98  222 

.46  554 

.53  446 

.01  778 

.55  224 

43 

18 

.44  819 

.98  218 

.46  601 

.53  399 

.01  782 

.55  181 

42 

19 

.44  862 

.98  215 

.46  648 

.53  352 

.01  785 

.55  138 

41 

20 

9.44  905 

9.98  211 

9.46  694 

0.53  306 

0.01  789 

0.55  095 

40 

21 

.44  948 

.98  207 

.46  741 

.53  259 

.01  793 

.55  052 

39 

22 

.44  992 

.98  204 

.46  788 

.53  212 

.01  796 

.55  008 

38 

23 

.45  035 

.98  200 

.46  835 

.53  165 

.01  800 

.54  965 

37 

24 

.45  077 

.98  196 

.46  881 

.53  119 

.01  804 

.54  923 

36 

25 

9.45  120 

9.98  192 

9.46  928 

0.53  072 

0.01  808 

0.54  880 

35 

26 

.45  163 

.98  189 

.46  975 

.53  025 

.01  811 

.54  837 

34 

27 

.45  206 

.98  185 

.47  021 

.52  979 

.01  815 

.54  794 

33 

28 

.45  249 

.98  181 

.47  068 

.52  932 

.01  819 

.54  751 

32 

29 

.45  292 

.98  177 

.47  114 

.52  886 

.01  823 

.54  708 

31 

30 

9.45  334 

9.98  174 

9.47  160 

0.52  840 

0.01  826 

0.54  666 

30 

31 

.45  377 

.98  170 

.47  207 

.52  793 

.01  830 

.54  623 

29 

32 

.45  419 

.98  166 

.47  253 

.52  747 

.01  834 

.54  581 

28 

33 

.45  462 

.98  162 

.47  299 

.52  701 

.01  838 

.54  538 

27 

34 

.45  504 

.98  159 

.47  346 

.52  654 

.01  841 

.54  496 

26 

35 

9.45  547 

9.98  155 

9.47  392 

0.52  608 

0.01  845 

0.54  453 

25 

36 

.45  589 

.98  151 

.47  438 

.52  562 

.01  849 

.54  411 

24 

37 

.45  632 

.98  147 

.47  484 

.52  516 

.01  853 

.54  368 

23 

38 

.45  674 

.98  144 

.47  530 

.52  470 

.01  856 

.54  326 

22 

39 

.45  716 

.98  140 

.47  576 

.52  424 

.01  860 

.54  284 

21 

40 

9.45  758 

9.98  136 

9.47  622 

0.52  378 

0.01  864 

0.54  242 

20 

41 

.45  801 

.98  132 

.47  668 

.52  332 

.01  868 

.54  199 

19 

42 

.45  843 

.98  129 

.47  714 

.52  286 

.01  871 

.54  157 

18 

43 

.45  885 

.98  125 

.47  760 

.52  240 

.01  875 

.54  115 

17 

44 

.45  927 

.98  121 

.47  806 

.52  194 

.01  879 

.54  073 

16 

45 

9.45  969 

9.98  117 

9.47  852 

0.52  148 

0.01  883 

0.54  031 

15 

46 

.46  011 

.98  113 

.  .47  897 

.52  103 

.01  887 

.53  989 

14 

47 

.46  053 

.98  110 

.47  943 

.52  057 

.01  890 

.53  947 

13 

48 

.46  095 

.98  106 

.47  989 

.52  011 

.01  894 

.53  905 

12 

49 

.46  136 

.98  102 

.48  035 

.51  965 

.01  898 

.53  864 

11 

50 

9.46  178 

9.98  098 

9.48  080 

0.51  920 

0.01  902 

0.53  822 

10 

51 

.46  220 

.98  094 

.48  126 

.51  874 

.01  906 

.53  780 

9 

52 

.46  262 

.98  090 

.48  171 

.51  829 

.01  910 

.53  738 

8 

53 

.46  303 

.98  087 

.48  217 

.51  783 

.01  913 

.53  697 

7 

54 

.46  345 

.98  083 

.48  262 

.51  738 

.01  917 

.53  655 

6 

55 

9.46  386 

9.98  079 

9.48  307 

0.51  693 

0.01  921 

0.53  614 

5 

56 

.46  428 

.98  075 

.48  353 

.51  647 

.01  925 

.53  572 

4 

57 

.46  469 

.98  071 

.48  398 

.51  602 

.01  929 

.53  531 

3 

58 

.46  511 

.98  067 

.48  443 

.51  557 

.01  933 

.53  489 

2 

59 

.46  552 

.98  063 

.48  489 

.51  511 

.01  937 

.53  448 

1 

60 

9.46  594 

9.98  060 

9.48  534 

0.51  466 

0.01  940 

0.53  406 

0 

Cos 

Sin 

Cot 

Tan 

Csc 

Sec 

/ 

106°  (26 

\6°) 

(253° 

)  73° 

Table  4.    Trigonometric  Logarithms 


213 


17°  (197°) 

(342°)  162° 

' 

Sin 

Cos 

Tan 

Cot 

Sec 

Csc 

0 

9.46  594 

9.98  060 

9.48  534 

0.51  466 

0.01  940 

0.53  406 

60 

1 

.46  635 

.98  056 

.48  579 

.51  421 

.01  944 

.53  365 

59 

2 

.46  676 

.98  052 

.48  624 

.51  376 

.01  948 

.53  324 

58 

3 

.46  717 

.98  048 

.  .48  669 

.51  331 

.01  952 

.53  283 

57 

4 

.46  758 

.98  044 

.48  714 

.51  286 

.01  956 

.53  242 

56 

5 

9.46  800 

9.98  040 

9.48  759 

0.51  241 

0.01  960 

0.53  200 

55 

6 

.46  841 

.98  036 

.48  804 

.51  196 

.01  964 

.53  159 

54 

7 

.46  882 

.98  032 

.48  849 

.51  151 

.01  968 

.53  118 

53 

8 

.46  923 

.98  029 

.48  894 

.51  106 

.01  971 

.53  077 

52 

9 

.46  964 

.98  025 

.48  939 

.51  061 

.01  975 

.53  036 

51 

10 

9.47  005 

9.98  021 

9.48  984 

0.51  016 

0.01  979 

0.52  995 

50 

11 

.47  045 

.98  017 

.49  029 

.50  971 

.01  983 

.52  955 

49 

12 

.47  086 

.98  013 

.49  073 

.50  927 

.01  987 

.52  914 

48 

13 

.47  127 

.98  009 

.49  118 

.50  882 

.01  991 

.52  873 

47 

14 

.47  168 

.98  005 

.49  163 

.50  837 

.01  995 

.52  832 

46 

15 

9.47  209 

9.98  001 

9.49  207 

0.50  793 

0.01  999 

0.52  791 

45 

16 

.47  249 

.97  997 

.49  252 

.50  748 

.02  003 

.52  751 

44 

17 

.47  290 

.97  993 

.49  296 

.50  704 

.02  007 

.52  710 

43 

18 

.47  330 

.97  989 

.49  341 

.50  659 

.02  Oil 

.52  670 

42 

19 

.47  371 

.97  986 

.49  385 

.50  615 

.02  014 

.52  629 

41 

20 

9.47  411 

9.97  982 

9.49  430 

0.50  570 

0.02  018 

0.52  589 

40 

21 

.47  452 

.97  978 

.49  474 

.50  526 

.02  022 

.52  548 

39 

22 

.47  492 

.97  974 

.49  519 

.50  481 

.02  026 

.52  508 

38 

23 

.47  533 

.97  970 

.49  563 

.50  437 

.02  030 

.52  467 

37 

24 

.47  573 

.97  966 

.49  607 

.50  393 

.02  034 

.52  427 

36 

25 

9.47  613 

9.97  962 

9.49  652 

0.50  348 

0.02  038 

0.52  387 

35 

26 

.47  654 

.97  958 

.49  696 

.50  304 

.02  042 

.52  346 

34 

27 

.47  694 

.97  954 

.49  740 

.50  260 

.02  046 

.52  306 

33 

28 

.47  734 

.97  950 

.49  784 

.50  216 

.02  050 

.52  266 

32 

29 

.47  774 

.97  946 

.49  828 

.50  172 

.02  054 

.52  226 

31 

30 

9.47  814 

9.97  942 

9.49  872 

0.50  128 

0.02  058 

0.52  186 

30 

31 

.47  854 

.97  938 

.49  916 

.50  084 

.02  062 

.52  146 

29 

32 

.47  894 

.97  934 

.49  960 

.50  040 

.02  066 

.52  106 

28 

33 

.47  934 

.97  930 

.50  004 

.49  996 

.02  070 

.52  066 

27 

34 

.47  974 

.97  926 

.50  048 

.49  952 

.02  074 

.52  026 

26 

35 

9.48  014 

9.97  922 

9.50  092 

0.49  908 

0.02  078 

0.51  986 

25 

36 

.48  054 

.97  918 

.50  136 

.49  864 

.02  082 

.51  946 

24 

37 

.48  094 

.97  914 

.50  180 

.49  820 

.02  086 

.51  906 

23 

38 

.48  133 

.97  910 

.50  223 

.49  777 

.02  090 

.51  867 

22 

39 

.48  173 

.97  906 

.50  207 

.49  733 

.02  094 

.51  827 

21 

40 

9.48  213 

9.97  902 

9.50  311 

0.49  689 

0.02  098 

0.51  787 

20 

41 

.48  252 

.97  898 

.50  355 

.49  645 

.02  102 

.51  748 

19 

42 

.48  292 

.97  894 

.50  398 

.49  602 

.02  106 

.51  708 

18 

43 

.48  332 

.97  890 

.50  442 

.49  558 

.02  110 

.51  668 

17 

44 

.48  371 

.97  886 

.50  485 

.49  515 

.02  114 

.51  629 

16 

45 

9.48  411 

9.97  882 

9.50  529 

0.49  471 

0.02  118 

0.51  589 

15 

46 

.48  450 

.97  878 

.50  572 

.49  428 

.02  122 

.51  550 

14 

47 

.48  490 

.97  874 

.50  616 

.49  384 

.02  126 

.51  510 

13 

48 

.48  529 

.97  870 

.50  659 

.49  341 

.02  130 

.51  471 

12 

49 

.48  568 

.97  866 

.50  703 

.49  297 

.02  134 

.51  432 

11 

50 

9.48  607 

9.97  861 

9.50  746 

0.49  254 

0.02  139 

0.51  393 

10 

51 

.48  647 

.97  857 

.50  789 

.49  211 

.02  143 

.51  353 

9 

52 

.48  686 

.97  853 

.50  833 

.49  167 

.02  147 

.51  314 

8 

53 

.48  725 

.97  849 

.50  876 

.49  124 

.02  151 

.51  275 

7 

54 

.48  764 

.97  845 

.50  919 

.49  081 

.02  155 

.51  236 

6 

55 

9.48  803 

9.97  841 

9.50  962 

0.49  038 

0.02  159 

0.51  197 

5 

56 

.48  842 

.97  837 

.51  005 

.48  995 

.02  163 

.51  158 

4 

57 

.48  881 

.97  833 

.51  048 

.48  952 

.02  167 

.51  119 

3 

58 

.48  920 

.97  829 

.51  092 

.48  908 

.02  171 

.51  080 

2 

59 

.48  959 

.97  825 

.51  135 

.48  865 

.02  175 

.51  041 

1 

60 

9.48  998 

9.97  821 

9.51  178 

0.48  822 

0.02  179 

0.51  002 
Sec 

0 

Cos 

Sin 

Cot 

Tan 

Csc 

t 

107°  (287°) 


(252°)  72= 


214 


Table  4.    Trigonometric  Logarithms 


18°  (198°) 

(341°)  161° 

/ 

Sin 

Cos 

Tan 

Cot 

Sec 

Csc 

0 

9.48  998 

9.97  821 

9.51  178 

0.48  822 

0.02  179 

0.51  002 

60 

1 

.49  037 

.97  817 

.51  221 

.48  779 

.02  183 

.50  963 

59 

2 

.49  076 

.97  812 

.51  264 

.48  736 

.02  188 

.50  924 

58 

3 

.49  115 

.97  808 

.51  306 

.48  694 

.02  192 

.50  885 

57 

4 

.49  153 

.97  804 

.51  349 

.48  651 

.02  196 

.50  847 

56 

5 

9.49  192 

9.97  800 

9.51  392 

0.48  608 

0.02  200 

0.50  808 

55 

6 

.49  231 

.97  796 

.51  435 

.48  565 

.02  204 

.50  769 

54 

7 

.49  269 

.97  792 

.51  478 

.48  522 

.02  208 

.50  731 

53 

8 

.49  308 

.97  788 

.51  520 

.48  480 

.02  212 

.50  692 

52 

9 

.49  347 

.97  784 

.51  563 

.48  437 

.02  216 

.50  653 

51 

10 

9.49  385 

9.97  779 

9.51  606 

0.48  394 

0.02  221 

0.50  615 

50 

11 

.49  424 

.97  775 

.51  648 

.48  352 

.02  225 

.50  576 

49 

12 

.49  462 

.97  771 

.51  691 

.48  309 

.02  229 

.50  538 

48 

13 

.49  500 

.97  767 

.51  734 

.48  266 

.02  233 

.50  500 

47 

14 

.49  539 

.97  763 

.51  776 

.48  224 

.02  237 

.50  461 

46 

15 

9.49  577 

9.97  759 

9.51  819 

0.48  181 

0.02  241 

0.50  423 

45 

16 

.49  615 

.97  754 

.51  861 

.48  139 

.02  246 

.50  385 

44 

17 

.49  654 

.97  750 

.51  903 

.48  097 

.02  250 

.50  346 

43 

18 

.49  692 

.97  746 

.51  946 

.48  054 

.02  254 

.50  308 

42 

19 

.49  730 

.97  742 

.51  988 

.48  012 

.02  258 

.50  270 

41 

20 

9.49  768 

9.97  738 

9.52  031 

0.47  969 

0.02  262 

0.50  232 

40 

21 

.49  806 

.97  734 

.52  073 

.47  927 

.02  266 

.50  194 

39 

22 

.49  844 

.97  729 

.52  115 

.47  885 

.02  271 

.50  156 

38 

23 

.49  882 

.97  725 

.52  157 

.47  843 

.02  275 

.50  118 

37 

24 

.49  920 

.97  721 

.52  200 

.47  800 

.02  279 

.50  080 

36 

25 

9.49  958 

9.97  717 

9.52  242 

0.47  758 

0.02  283 

0.50  042 

35 

26 

.49  996 

.97  713 

.52  284 

.47  716 

.02  287 

.50  004 

34 

27 

.50  034 

.97  708 

.52  326 

.47  674 

.02  292 

.49  966 

33 

28 

.50  072 

.97  704 

.52  368 

.47  632 

.02  296 

.49  928 

32 

29 

.50  110 

.97  700 

.52  410 

.47  590 

.02  300 

.49  890 

31 

30 

9.50  148 

9.97  696 

9.52  452 

0.47  548 

0.02  304 

0.49  852 

30 

31 

.50  185 

.97  691 

.52  494 

.47  506 

.02  309 

.49  815 

29 

32 

.50  223 

.97  687 

.52  536 

.47  464 

.02  313 

.49  777 

28 

33 

.50  261 

.97  683 

.52  578 

.47  422 

.02  317 

.49  739 

27 

34 

.50  298 

.97  679 

.52  620 

.47  380 

.02  321 

.49  702 

26 

35 

9.50  336 

9.97  674 

9.52  661 

0.47  339 

0.02  326 

0.49  664 

25 

36 

.50  374 

.97  670 

.52  703 

.47  297 

.02  330 

.49  626 

24 

37 

.50  411 

.97  666 

.52  745 

.47  255 

.02  334 

.49  589 

23 

38 

.50  449 

.97  662 

.52  787 

.47  213 

.02  338 

.49  551 

22 

39 

.50  486 

.97  657 

.52  829 

.47  171 

.02  343 

.49  514 

21 

40 

9.50  523 

9.97  653 

9.52  870 

0.47  130 

0.02  347 

0.49  477 

20 

41 

.50  561 

.97  649 

.52  912 

.47  088 

.02  351 

.49  439 

19 

42 

.50  598 

.97  645 

.52  953 

.47  047 

.02  355 

.49  402 

18 

43 

.50  635 

.97  640 

.52  995 

.47  005 

.02  360 

.49  365 

17 

44 

.50  673 

.97  636 

.53  037 

.46  963 

.02  364 

.49  327 

16 

45 

9.50  710 

9.97  632 

9.53  078 

0.46  922 

0.02  368 

0.49  290 

15 

46 

.50  747 

.97  628 

.53  120 

.46  880 

.02  372 

.49  253 

14 

47 

.50  784 

.97  623 

.53  161 

.46  839 

.02  377 

.49  216 

13 

48 

.50  821 

.97  619 

.53  202 

.46  798 

.02  381 

.49  179 

12 

49 

.50  858 

.97  615 

.53  244 

.46  756 

.02  385 

.49  142 

11 

50 

9.50  896 

9.97  610 

9.53  285 

0.46  715 

0.02  390 

0.49  104 

10 

51 

.50  933 

.97  606 

.53  327 

.46  673 

.02  394 

.49  067 

9 

52 

.50  970 

.97  602 

.53  368 

.46  632 

.02  398 

.49  030 

8 

53 

.51  007 

.97  597 

.53  409 

.46  591 

.02  403 

.48  993 

7 

54 

.51  043 

.97  593 

.53  450 

.46  550 

.02  407 

.48  957 

6 

55 

9.51  080 

9.97  589 

9.53  492 

0.46  508 

0.02  411 

0.48  920 

5 

56 

.51  117 

.97  584 

.53  533 

.46  467 

.02  416 

.48  883 

4 

57 

.51  154 

.97  580 

.53  574 

.46  426 

.02  420 

.48  846 

3 

58 

.51  191 

.97  576 

.53  615 

.46  385 

.02  424 

.48  809 

2 

59 

.51  227 

.97  571 

.53  656 

.46  344 

.02  429 

.48  773 

1 

60 

9.51  264 

9.97  567 

9.53  697 

0.46  303 

0.02  433 

0.48  736 

0 

Cos 

Sin 

Cot 

Tan 

Csc 

Sec 

/ 

108°  (288°) 


(251°)  71" 


Table  4.    Trigonometric  Logarithms 


215 


19°  (199°) 

(340°)  160° 

/ 

Sin 

Cos 

Tan 

Cot 

Sec 

Csc 

0 

9.51  264 

9.97  567 

9.53  697 

0.46  303 

0.02  433 

0.48  736 

60 

1 

.51  301 

.97  563 

.53  738 

.46  262 

.02  437 

.48  699 

59 

2 

.51  338 

.97  558 

.53  779 

.46  221 

.02  442 

.48  662 

58 

3 

.51  374 

.97  554 

.53  820 

.46  180 

.02  446 

.48  626 

57 

4 

.51  411 

.97  550 

.53  861 

.46  139 

.02  450 

.48  589 

56 

5 

9.51  447 

9.97  545 

9.53  902 

0.46  098 

0.02  455 

0.48  553 

55 

6 

.51  484 

.97  541 

.53  943 

.46  057 

.02  459 

.48  516 

54 

7 

.51  520 

.97  536 

.53  984 

.46  016 

.02  464 

.48  480 

53 

8 

.51  557 

.97  532 

.54  025 

.45  975 

.02  468 

.48  443 

52 

9 

.51  593 

.97  528 

.54  065 

.45  935 

.02  472 

.48  407 

51 

10 

9.51  629 

9.97  523 

9.54  106 

0.45  894 

0.02  477 

0.48  371 

50 

11 

.51  666 

.97  519 

.54  147 

.45  853 

.02  481 

.48  334 

49 

12 

.51  702 

.97  515 

.54  187 

.45  813 

.02  485 

.48  298 

48 

13 

.51  738 

.97  510 

.54  228 

.45  772 

.02  490 

.48  262 

47 

14 

.51  774 

.97  506 

.54  269 

.45  731 

.02  494 

.48  226 

46 

15 

9.51  811 

9.97  501 

9.54  309 

0.45  691 

0.02  499 

0.48  189 

45 

16 

.51  847 

.97  497 

.54  350 

.45  650 

.02  503 

.48  153 

44 

17 

.51  883 

.97  492 

.54  390 

.45  610 

.02  508 

.48  117 

43 

18 

.51  919 

.97  488 

.54  431 

.45  569 

.02  512 

.48  081 

42 

19 

.51  955 

.97  484 

.54  471 

.45  529 

.02  516 

.48  045 

41 

20 

9.51  991 

9.97  479 

9.54  512 

0.45  488 

0.02  521 

0.48  009 

40 

21 

.52  027 

.97  475 

.54  552 

.45  448 

.02  525 

.47  973 

39 

22 

.52  063 

.97  470 

.54  593 

.45  407 

.02  530 

.47  937 

38 

23 

.52  099 

.97  466 

.54  633 

.45  367 

.02  534 

.47  901 

37 

24 

.52  135 

.97  461 

.54  673 

.45  327 

.02  539 

.47  865 

36 

25 

9.52  171 

9.97  457 

9.54  714 

0.45  286 

0.02  543 

0.47  829 

35 

26 

.52  207 

.97  453 

.54  754 

.45  246 

.02  547 

.47  793 

34 

27 

.52  242 

.97  448 

.54  794 

.45  206 

.02  552 

.47  758 

33 

28 

.52  278 

.97  444 

.54  835 

.45  165 

.02  556 

.47  722 

32 

29 

.52  31,4 

.97  439 

.54  875 

.45  125 

.02  561 

.47  686 

31 

30 

9.52  350 

9.97  435 

9.54  915 

0.45  085 

0.02  565 

0.47  650 

30 

31 

.52  385 

.97  430 

.54  955 

.45  045 

.02  570 

.47  615 

29 

32 

.52  421 

.97  426 

.54  995 

.45  005 

.02  574 

.47  579 

28 

33 

.52  456 

.97  421 

.55  035 

.44  965 

.02  579 

.47  544 

27 

34 

.52  492 

.97  417 

.55  075 

.44  925 

.02  583 

.47  508 

26 

35 

9.52  527 

9.97  412 

9.55  115 

0.44  885 

0.02  588 

0.47  473 

25 

36 

.52  563 

.97  408 

.55  155 

.44  845 

.02  592 

.47  437 

24 

37 

.52  598 

.97  403 

.55  195 

.44  805 

.02  597 

.47  402 

23 

38 

.52  634 

.97  399 

.55  235 

.44  765 

.02  601 

.47  366 

22 

39 

.52  669 

.97  394 

.55  275 

.44  725 

.02  606 

.47  331 

21 

40 

9.52  705 

9.97  390 

9.55  315 

0.44  685 

0.02  610 

0.47  295 

20 

41 

•  .52  740 

.97  385 

.55  355 

.44  645 

.02  615 

.47  260 

19 

42 

.52  775 

.97  381 

.55  395 

.44  605 

.02  619 

.47  225 

18 

43 

.52  811 

.97  376 

.55  434 

.44  566 

.02  624 

.47  189 

17 

44 

.52  846 

.97  372 

.55  474 

.44  526 

.02  628 

.47  154 

16 

45 

9.52  881 

9.97  367 

9.55  514 

0.44  486 

0.02  633 

0.47  119 

15 

46 

.52  916 

.97  363 

.55  554 

.44  446 

.02  637 

.47  084 

14 

47 

.52  951 

.97  358 

.55  593 

.44  407 

.02  642 

.47  049 

13 

48 

.52  986 

.97  353 

.55  633 

.44  367 

.02  647 

.47  014 

12 

49 

.53  021 

.97  349 

.55  673 

.44  327 

.02  651 

.46  979 

11 

50 

9.53  056 

9.97  344 

9.55  712 

0.44  288 

0.02  656 

0.46  944 

10 

51 

.53  092 

.97  340 

.55  752 

.44  248 

.02  660 

.46  908 

9 

52 

.53  126 

.97  335 

.55  791 

.44  209 

.02  665 

.46  874 

8 

53 

.53  161 

.97  331 

.55  831 

.44  169 

.02  669 

.46  839 

7 

54 

.53  196 

.97  326 

.55  870 

.44  130 

.02  674 

.46  804 

6 

55 

9.53  231 

9.97  322 

9.55  910 

0.44  090 

0.02  678 

0.46  769 

5 

56 

.53  266 

.97  317 

.55  949 

.44  051 

.02  683 

.46  734 

4 

57 

.53  301 

.97  312 

.55  989 

.44  011 

.02  688 

.46  699 

3 

58 

.53  336 

.97  308 

.56  028 

.43  972 

.02  692 

.46  664 

2 

59 

.53  370 

.97  303 

.56  067 

.43  933 

.02  697 

.46  630 

1 

60 

9.53  405 

9.97  299 

9.56  107 

9.43  893 

0.02  701 

0.46  595 

0 

Cos 

Sin 

Cot 

Tan 

Csc 

Sec 

/ 

109°  (289°) 


(250°)  70= 


216 


Table  4.    Trigonometric  Logarithms 


20°  (200°) 

(339°)  159° 

/ 

Sin 

Cos 

Tan 

Cot 

Sec 

Csc 

0 

9.53  405 

9.97  299 

9.56  107 

0.43  893 

0.02  701 

0.46  595 

60 

1 

.53  440 

.97  294 

.56  146 

.43  854 

.02  706 

.46  560 

59 

2 

.53  475 

.97  289 

.56  185 

.43  815 

.02  711 

.46  525 

58 

3 

.53  509 

.97  285 

.56  224 

.43  776 

.02  715 

.46  491 

57 

4 

.53  544 

.97  280 

.56  264 

.43  736 

.02  720 

.46  456 

56 

5 

9.53  578 

9.97  276 

9.56  303 

0.43  697 

0.02  724 

0.46  422 

55 

6 

.53  613 

.97  271 

.56  342 

.43  658 

.02  729 

.46  387 

54 

7 

.53  647 

.97  266 

.56  381 

.43  619 

.02  734 

.46  353 

53 

8 

.53  682 

.97  262 

.56  420 

.43  580 

.02  738 

.46  318 

52 

9 

.53  716 

.97  257 

.56  459 

.43  541 

.02  743 

.46  284 

51 

10 

9.53  751 

9.97  252 

9.56  498 

0.43  502 

0.02  748 

0.46  249 

50 

11 

.53  785 

.97  248 

.56  537 

.43  463 

.02  752 

.46  215 

49 

12 

.53  819 

.97  243 

.56  576 

.43  424 

.02  757 

.46  181 

48 

13 

.53  854 

.97  238 

.56  615 

.43  385 

.02  762 

.46  146 

47 

14 

.53  888 

.97  234 

.56  654 

.43  346 

.02  766 

.46  112 

46 

15 

9.53  922 

9.97  229 

9.56  693 

0.43  307 

0.02  771 

0.46  078 

45 

16 

.53  957 

.97  224 

.56  732 

.43  268 

.02  776 

.46  043 

44 

17 

.53  991 

.97  220 

.56  771 

.43  229 

.02  780 

.46  009 

43 

18 

.54  025 

.  .97  215 

.56  810 

.43  190 

.02  785 

.45  975 

42 

19 

.54  059 

.97  210 

.56  849 

.43  151 

.02  790 

.45  941 

41 

20 

9.54  093 

9.97  206 

9.56  887 

0.43  113 

0.02  794 

0.45  907 

40 

21 

.54  127 

.97  201 

.56  926 

.43  074 

.02  799 

.45  873 

39 

22 

.54  161 

.97  196 

.56  965 

.43  035 

.02  804 

.45  839 

38 

23 

.54  195 

.97  192 

.57  004 

.42  996 

.02  808 

.45  805 

37 

24 

.54  229 

.97  187 

.57  042 

.42  958 

.02  813 

.45  771 

36 

25 

9.54  263 

9.97  182 

9.57  081 

0.42  919 

0.02  818 

0.45  737 

35 

26 

.54  297 

.97  178 

.57  120 

.42  880 

.02  822 

.45  703 

34 

27 

.54  331 

.97  173 

.57  158 

.42  842 

.02  827 

.45  669 

33 

28 

.54  365 

.97  168 

.57  197 

.42  803 

.02  832 

.45  635 

32 

29 

.54  399 

.97  163 

.57  235 

.42  765 

.02  837 

.45  601 

31 

30 

9.54  433 

9.97  159 

9.57  274 

0.42  726 

0.02  841 

0.45  567 

30 

31 

.54  466 

.97  154 

.57  312 

.42  688 

.02  846 

.45  534 

29 

32 

.54  500 

.97  149 

.57  351 

.42  649 

.02  851 

.45  500 

28 

33 

.54  534 

.97  145 

.57  389 

.42  611 

.02  855 

.45  466 

27 

34 

.54  567 

.97  140 

.57  428 

.42  572 

.02  860 

.45  433 

26 

35 

9.54  601 

9.97  135 

9.57  466 

0.42  534 

0.02  865 

0.45  399 

25 

36 

.54  635 

.97  130 

.57  504 

.42  496 

.02  870 

.45  365 

24 

37 

.54  668 

.97  126 

.57  543 

.42  457 

.02  874 

.45  332 

23 

38 

.54  702 

.97  121 

.57  581 

.42  419 

.02  879 

.45  298 

22 

39 

.54  735 

.97  116 

.57  619 

.42  381 

.02  884 

.45  265 

21 

40 

9.54  769 

9.97  111 

9.57  658 

0.42  342 

0.02  889 

0.45  231 

20 

41 

.54  802 

.97  107 

.57  696 

.42  304 

.02  893 

.45  198 

19 

42 

.54  836 

.97  102 

.57  734 

.42  266 

.02  898 

.45  164 

18 

43 

.54  869 

.97  097 

.57  772 

.42  228 

.02  903 

.45  131 

17 

44 

.54  903 

.97  092 

.57  810 

.42  190 

.02  908 

.45  097 

16 

45 

9.54  936 

9.97  087 

9.57  849 

0.42  151 

0.02  913 

0.45  064 

15 

46 

.54  969 

.97  083 

;57  887 

.42  113 

.02  917 

.45  031 

14 

47 

.55  003 

.97  078 

.57  925 

.42  075 

.02  922 

.44  997 

13 

48 

.55  036 

.97  073 

.57  963 

.42  037 

.02  927 

.44  964 

12 

49 

.55  069 

.97  068 

.58  001 

.41  999 

.02  932 

.44  931 

11 

50 

9.55  102 

9.97  063 

9.58  039 

0.41  961 

0.02  937 

0.44  898 

10 

51 

.55  136 

.97  059 

.58  077 

.41  923 

.02  941 

.44  864 

9 

52 

.55  169 

.97  054 

.58  115 

.41  885 

.02  946 

.44  831 

8 

53 

.55  202 

.97  049 

.58  153 

.41  847 

.02  951 

.44  798 

7 

54 

.55  235 

.97  044 

.58  191 

.41  809 

.02  956 

.44  765 

6 

55 

9.55  268 

9.97  039 

9.58  229 

0.41  771 

0.02  961 

0.44  732 

5 

56 

.55  301 

.97  035 

.58  267 

.41  733 

.02  965 

.44  699 

4 

57 

.55  334 

.97  030 

.58  304 

.41  696 

.02  970 

.44  666 

3 

58 

.  .55  367 

.97  025 

.58  342 

.41  658 

.02  975 

.44  633 

2 

59 

.55  400 

.97  020 

.58  380 

.41  620 

.02  980 

.44  600 

1 

60 

9.55  433 

9.97  015 

9.58  418 

0.41  582 

0.02  985 

0.44  567 
Sec 

0 

Cos 

Sin 

Cot 

Tan 

Csc 

1 

110°  (290°) 


(249°)  69= 


Table  4.    Trigonometric  Logarithms 


217 


21°  (201°) 

(338° 

)  158° 

/ 

Sin 

Cos 

Tan 

Cot 

Sec 

Csc 

0 

9.55  433 

9.97  015 

9.58  418 

0.41  582 

0.02  985 

0.44  567 

60 

1 

.55  466 

.97  010 

.58  455 

.41  545 

.02  990 

.44  534 

59 

2 

.55  499 

.97  005 

.58  493 

.41  507 

.02  995 

.44"  501 

58 

3 

.55  532 

.97  001 

.58  531 

.41  469 

.02  999 

.44  468 

57 

4 

.55  564 

.96  996 

.58  569 

.41  431 

.03  004 

.44  436 

56 

5 

9.55  597 

9.96  991 

9.58  606 

0.41  394 

0.03  009 

0.44  403 

55 

6 

.55  630 

.96  986 

.58  644 

.41  356 

.03  014 

.44  370 

54 

7 

.55  663 

.96  981 

.58  681 

.41  319 

.03  019 

.44  337 

53 

8 

.55  695 

.96  976 

.58  719 

.41  281 

.03  024 

.44  305 

52 

9 

.55  728 

.96  971 

.58  757 

.41  243 

.03  029 

.44  272 

51 

10 

9.55  761 

9.96  966 

9.58  794 

0.41  206 

0.03  034 

0.44  239 

50 

11 

.55  793 

.96  962 

.58  832 

.41  168 

.03  038 

.44  207 

49 

12 

.55  826 

.96  957 

.58  869 

.41  131 

.03  043 

.44  174 

48 

13 

.55  858 

.96  952 

.58  907 

.41  093 

.03  048 

.44  142 

47 

14 

.55  891 

.96  947 

.58  944 

.41  056 

.03  053 

.44  109 

46 

15 

9.55  923 

9.96  942 

9.58  981 

0.41  019 

0.03  058 

0.44  077 

45 

16 

.55  956 

.96  937 

.59  019 

.40  981 

.03  063 

.44  044 

44 

17 

.55  988 

.96  932 

.59  056 

.40  944 

.03  068 

.44  012 

43 

18 

.56  021 

.96  927 

.59  094 

.40  906 

.03  073 

.43  979 

42 

19 

.56  053 

.96  922 

.59  131 

.40  869 

.03  078 

.43  947 

41 

20 

9.56  085 

9.96  917 

9.59  168 

0.40  832 

0.03  083 

0.43  915 

40 

21 

.56  118 

.96  912 

.59  205 

.40  795 

.03  088 

.43  882 

39 

22 

.56  150 

.96  907 

.59  243 

.40  757 

.03  093 

.43  850 

38 

23 

.56  182 

.96  903 

.59  280 

.40  720 

.03  097 

.43  818 

37 

24 

.56  215 

.96  898 

.59  317 

.40  683 

.03  102 

.43  785 

36 

25 

9.56  247 

9.96  893 

9.59  354 

0.40  646 

0.03  107 

0.43  753 

35 

26 

.56  279 

.96  888 

.59  391 

.40  609 

.03  112 

.43  721 

34 

27 

.56  311 

.96  883 

.59  429 

.40  571 

.03  117 

.43  689 

33 

28 

.56  343 

.96  878 

.59  466 

.40  534 

.03  122 

.43  657 

32 

29 

.56  375 

.96  873 

.59  503 

.40  497 

.03  127 

.43  625 

31 

30 

9.56  408 

9.96  868 

9.59  540 

0.40  460 

0.03  132 

0.43  592 

30 

31 

.56  440 

.96  863 

.59  577 

.40  423 

.03  137 

.43  560 

29 

32 

.56  472 

.96  858 

.59  614 

.40  386 

.03  142 

.43  528 

28 

33 

.56  504 

.96  853 

.59  651 

.40  349 

.03  147 

.43  496 

27 

34 

.56  536 

.96  848 

.59  688 

.40  312 

.03  152 

.43  464 

26 

35 

9.56  568 

9.96  843 

9.59  725 

0.40  275 

0.03  157 

0.43  432 

25 

36 

.56  599 

.96  838 

.59  762 

.40  238 

.03  162 

.43  401 

24 

37 

.56  631 

.96  833 

.59  799 

.40  201 

.03  167 

.43  369 

23 

38 

.56  663 

.96  828 

.59  835 

.40  165 

.03  172 

.43  337 

22 

39 

.56  695 

.96  823 

.59  872 

.40  128 

.03  177 

.43  305 

21 

40 

9.56  727 

9.96  818 

9.59  909 

0.40  091 

0.03  182 

0.43  273 

20 

41 

.56  759 

.96  813 

.59  946 

.40  054 

.03  187 

.43  241 

19 

42 

.56  790 

.96  808 

.59  983 

.40  017 

.03  192 

.43  210 

18 

43 

•  .56  822 

.96  803 

.60  019 

.39  981 

.03  197 

.43  178 

17 

44 

.56  854 

.96  798 

.60  056 

.39  944 

.03  202 

.43  146 

16 

45 

9.56  886 

9.96  793 

9.60  093 

0.39  907 

0.03  207 

0.43  114 

15 

46 

.56  917 

.96  788 

.60  130 

.39  870 

.03  212 

.43  083 

14 

47 

.56  949 

.96  783 

.60  166 

.39  834 

.03  217 

.43  051 

13 

48 

.56  980 

.96  778 

.60  203 

.39  797 

.03  222 

.43  020 

12 

49 

.57  012 

.96  772 

.60  240 

.39  760 

.03  228 

.42  988 

11 

50 

9.57  044 

9.96  767 

9.60  276 

0.39  724 

0.03  233 

0.42  956 

10 

51 

.57  075 

.96  762 

.60  313 

.39  687 

.03  238 

.42  925 

9 

52 

.57  107 

.96  757 

.60  349 

.39  651 

.03  243 

.42  893 

8 

53 

.57  138 

.96  752 

.60  386 

.39  614 

.03  248 

.42  862 

7 

54 

.57  169 

.96  747 

.60  422 

.39  578 

.03  253 

.42  831 

6 

55 

9.57  201 

9.96  742 

9.60  459 

0.39  541 

0.03  258 

0.42  799 

5 

56 

.57  232 

.96  737 

.60  495 

.39  505 

.03  263 

.42  768 

4 

57 

.57  264 

.96  732 

.60  532 

.39  468 

.03  268 

.42  736 

3 

58 

.57  295 

.96  727 

.60  568 

.39  432 

.03  273 

.42  705 

2 

59 

.57  326 

.96  722 

.60  605 

.39  395 

.03  278 

.42  674 

1 

60 

9.57  358 

9.96  717 

9.60  641 

0.39  359 

0.03  283 

0.42  642 

0 

Cos 

Sin 

Cot 

Tan 

Csc 

Sec 

/ 

111°  (291°) 


(248°)  68"= 


218 


Table  4.    Trigonometric  Logarithms 


22°  (202°) 

(337°)  157° 

/ 

Sin 

Cos 

Tan 

Cot 

Sec 

Csc 

0 

9.57  358 

9.96  717 

9.60  641 

0.39  359 

0.03  283 

0.42  642 

60 

1 

.57  389 

.96  711 

.60  677 

.39  323 

.03  289 

.42  611 

59 

2 

.57  420 

.96  706 

.60  714 

.39  286 

.03  294 

.42  580 

58 

3 

.57  451 

.96  701 

.60  750 

.39  250 

.03  299 

.42  549 

57 

4 

.57  482 

.96  696 

.60  786 

.39  214 

.03  304 

.42  518 

56 

5 

9.57  514 

9.96  691 

9.60  823 

0.39  177 

0.03  309 

0.42  486 

55 

6 

.57  545 

.96  686 

.60  859 

.39  141 

.03  314 

.42  455 

54 

7 

.57  576 

.96  681 

.60  895 

.39  105 

.03  319 

.42  424 

53 

8 

.57  607 

.96  676 

.60  931 

.39  069 

.03  324 

.42  393 

52 

9 

.57  638 

.96  670 

.60  967 

.39  033 

.03  330 

.42  362 

51 

10 

9.57  669 

9.96  665 

9.61  004 

9.38  996 

0.03  335 

0.42  331 

50 

11 

.57  700 

.96  660 

.61  040 

.38  960 

.03  340 

.42  300 

49 

12 

.57  731 

.96  655 

.61  076 

.38  924 

.03  345 

.42  269 

48 

13 

.57  762 

.96  650 

.61  112 

.38  888 

.03  350 

.42  238 

47 

14 

.57  793 

.96  645 

,61  148 

.38  852 

.03  355 

.42  207 

46 

15 

9.57  824 

9.96  640 

9.61  184 

0.38  816 

0.03  360 

0.42  176 

45 

16 

.57  855 

.96  634 

.61  220 

.38  780 

.03  366 

.42  145 

44 

17 

.57  885 

.96  629 

.61  256 

.38  744 

.03  371 

.42  115 

43 

18 

.57  916 

.96  624 

.61  292, 

.38  708 

.03  376 

.42  084 

42 

19 

.57  947 

.96  619 

.61  328 

.38  672 

.03  381 

.42  053 

41 

20 

9.57  978 

9.96  614 

9.61  364 

0.38  636 

0.03  386 

0.42  022 

40 

21 

.58  008 

.96  608 

.61  400 

.38  600 

.03  392 

.41  992 

39 

22 

.58  039 

.96  603 

.61  436 

.38  564 

.03  397 

.41  961 

38 

23 

.58  070 

.96  598 

.61  472 

.38  528 

.03  402 

.41  930 

37 

24 

.58  101 

.96  593 

.61  508 

.38  492 

.03  407 

.41  899 

36 

25 

9.58  131 

9.96  588 

9.61  544 

0.38  456 

0.03  412 

0.41  869 

35 

26 

.58  162 

.96  582 

.61  579 

.38  421 

.03  418 

.41  838 

34 

27 

.58  192 

.96  577 

.61  615 

.38  385 

.03  423 

.41  808 

33 

28 

.58  223 

.96  572 

.61  651 

.38  349 

.03  428 

.41  777 

32 

29 

.58  253 

.96  567 

.61  687 

.38  313 

.03  433 

.41  747 

31 

30 

9.58  284 

9.96  562 

9.61  722 

0.38  278 

0.03  438 

0.41  716 

30 

31 

.58  314 

.96  556 

.61  758 

.38  242 

.03  444 

.41  686 

29 

32 

.58  345 

.96  551 

.61  794 

.38  206 

.03  449 

.41  655 

28 

33 

.58  375 

.96  546 

.61  830 

.38  170 

.03  454 

.41  625 

27 

34 

.58  406 

.96  541 

.61  865 

.38  135 

.03  459 

.41  594 

26 

35 

9.58  436 

9.96  535 

9.61  901 

0.38  099 

0.03  465 

0.41  564 

25 

36 

.58  467 

.96  530 

.61  936 

.38  064 

.03  470 

.41  533 

24 

37 

.58  497 

.96  525 

.61  972 

.38  028 

.03  475 

.41  503 

23 

38 

.58  527 

.96  520 

.62  008 

.37  992 

.03  480 

.41  473 

22 

39 

.58  557 

.96  514 

.62  043 

.37  957 

.03  486 

.41  443 

21 

40 

9.58  588 

9.96  509 

9.62  079 

0.37  921 

0.03  491 

0.41  412 

20 

41 

.58  618 

.96  504 

.62  114 

.37  886 

.03  496 

.41  382 

19 

42 

.58  648 

.96  498 

.62  150 

.37  850 

.03  502 

.41  352 

18 

43 

.58  678 

.96  493 

.62  185 

.37  815 

.03  507 

.41  322 

17 

44 

.58  709 

.96  488 

.62  221 

.37  779 

.03  512 

.41  291 

16 

45 

9.58  739 

9.96  483 

9.62  256 

0.37  744 

0.03  517 

0.41  261 

15 

46 

.58  769 

.96  477 

.62  292 

.37  708 

.03  523 

.41  231 

14 

47 

.58  799 

.96  472 

.62  327 

.37  673 

.03  528 

.41  201 

13 

48 

.58  829 

.96  467 

.62  362 

.37  638 

.03  533 

.41  171 

12 

49 

.58  859 

.96  461 

.62  398 

.37  602 

.03  539 

.41  141 

11 

50 

9.58  889 

9.96  456 

9.62  433 

0.37  567 

0.03  544 

0.41  111 

10 

51 

.58  919 

.96  451 

.62  468 

.37  532 

.03  549 

.41  081 

9 

52 

.58  949 

.96  445 

.62  504 

.37  496 

.03  555 

.41  051 

8 

53 

.58  979 

.96  440 

.62  539 

.37  461 

.03  560 

.41  021 

7 

54 

.59  009 

.96  435 

.62  574 

.37  426 

.03  565 

.40  991 

6 

55 

9.59  039 

9.96  429 

9.62  609 

0.37  391 

0.03  571 

0.40  961 

5 

56 

.59  069 

.96  424 

.62  645 

.37  355 

.03  576 

.40  931 

4 

57 

.59  098 

.96  419 

.62  680 

.37  320 

.03  581 

.40  902 

3 

58 

.59  128 

.96  413 

.62  715 

.37  285 

.03  587 

.40  872 

2 

59 

.59  158 

.96  408 

.62  750 

.37  250 

.03  592 

.40  842 

1 

60 

9.59  188 
Cos 

9.96  403 

9.62  785 

0.37  215 

0.03  597 

0.40  812 

0 

Sin 

Cot 

Tan 

Csc 

Sec 

/ 

112'  (292°) 


(247°)  67' 


Table  4.    Trigonometric  Logarithms 


219 


23°  (203°) 

• 

(336°)  156° 

/ 

Sin 

Cos 

Tan 

Cot 

Sec 

Csc 

0 

9.59  188 

9.96  403 

9.62  785 

0.37  215 

0.03  597 

0.40  812 

60 

1 

.59  218 

.96  397 

.62  820 

.37  180 

.03  603 

.40  782 

59 

2 

.59  247 

.96  392 

.62  855 

.37  145 

.03  608 

.40  753 

58 

3 

.59  277 

.96  387 

.62  890 

.37  110 

.03  613 

.40  723 

57 

4 

.59  307 

.96  381 

.62  926 

.37  074 

.03  619 

.40  693 

56 

5 

9.59  336 

9.96  376 

9.62  961 

0.37  039 

0.03  624 

0.40  664 

55 

6 

.59  366 

.96  370 

.62  996 

.37  004 

.03  630 

.40  634 

54 

7 

.59  396 

.96  365 

.63  031 

.36  969 

.03  635 

.40  604 

53 

8 

.59  425 

.96  360 

.63  066 

.36  934 

.03  640 

.40  575 

52 

9 

.59  455 

.96  354 

.63  101 

.36  899 

.03  646 

.40  545 

51 

10 

9.59  484 

9.96  349 

9.63  135 

0.36  865 

0.03  651 

0.40  516 

50 

11 

.59  514 

.96  343 

.63  170 

.36  830 

.03  657 

.40  486 

49 

12 

.59  543 

.96  338 

.63  205 

.36  795 

.03  662 

.40  457 

48 

13 

.59  573 

.96  333 

.63  240 

.36  760 

.03  667 

.40  427 

47 

14 

.59  602 

.96  327 

.63  275 

.36  725 

.03  673 

.40  398 

46 

15 

9.59  632 

9.96  322 

9.63  310 

0.36  690 

0.03  678 

0.40  368 

45 

16 

.59  661 

.96  316 

.63  345 

.36  655 

.03  684 

.40  339 

44 

17 

.59  690 

.96  311 

.63  379 

.36  621 

.03  689 

.40  310 

43 

18 

.59  720 

.96  305 

.63  414 

.36  586 

.03  695 

.40  280 

42 

19 

.59  749 

.96  300 

.63  449 

.36  551 

.03  700 

.40  251 

41 

20 

9.59  778 

9.96  294 

9.63  484 

0.36  516 

0.03  706 

0.40  222 

40 

21 

.59  808 

.96  289 

.63  519 

.36  481 

.03  711 

.40  192 

39 

22 

.59  837 

.96  284 

.63  553 

.36  447 

.03  716 

.40  163 

38 

23 

.59  866 

.96  278 

.63  588 

.36  412 

.03  722 

.40  134 

37 

24 

.59  895 

.96  273 

.63  623 

.36  377 

.03  727 

.40  105 

36 

25 

9.59  924 

9.96  267 

9.63  657 

0.36  343 

0.03  733 

0.40  076 

35 

26 

.59  954 

.96  262 

.63  692 

.36  308 

.03  738 

.40  046 

34 

27 

.59  983 

.96  256 

.63  726 

.36  274 

.03  744 

.40  017 

33 

28 

.60  012 

.96  251 

.63  761 

.36  239 

.03  749 

.39  988 

32 

29 

.60  041 

.96  245 

.63  796 

.36  204 

.03  755 

.39  959 

31 

30 

9.60  070 

9.96  240 

9.63  830 

0.36  170 

0.03  760 

.39  930 

30 

31 

.60  099 

.96  234 

.63  865 

.36  135 

.03  766 

.39  901 

29 

32 

.60  128 

.96  229 

.63  899 

.36  101 

.03  771 

.39  872 

28 

33 

.60  157 

.96  223 

.63  934 

.36  066 

.03  777 

.39  843 

27 

34 

.60  186 

.96  218 

.63  968 

.36  032 

.03  782 

.39  814 

26 

35 

9.60  215 

9.96  212 

9.64  003 

0.35  997 

0.03  788 

0.39  785 

25 

36 

.60  244 

.96  207 

.64  037 

.35  963 

.03  793 

.39  756 

24 

37 

.60  273 

.96  201 

.64  072 

.35  928 

.03  799 

.39  727 

23 

38 

.60  302 

.96  196 

.64  106 

.35  894 

.03  804 

.39  698 

22 

39 

.60  331 

.96  190 

.64  140 

.35  860 

.03  810 

.39  669 

21 

40 

9.60  359 

9.96  185 

9.64  175 

0.35  825 

0.03  815 

0.39  641 

20 

41 

.60  388 

.96  179 

.64  209 

.35  791 

.03  821 

.39  612 

19 

42 

.60  417 

.96  174 

.64  243 

.35  757 

.03  826 

.39  583 

18 

43 

.60  446 

.96  168 

.64  278 

.35  722 

.03  832 

.39  554 

17 

44 

.60  474 

.96  162 

.64  312 

.35  688 

.03  838 

.39  526 

16 

45 

9.60  503 

9.96  157 

9.64  346 

0.35  654 

0.03  843 

0.39  497 

15 

46 

.60  532 

.96  151 

.64  381 

.35  619 

.03  849 

.39  468 

14 

47 

.60  561 

.96  146 

.64  415 

.35  585 

.03  854 

.39  439 

13 

48 

.60  589 

.96  140 

.64  449 

.35  551 

.03  860 

.39  411 

12 

49 

.60  618 

.96  135 

.64  483 

.35  517 

.03  865 

.39  382 

11 

50 

9.60  646 

9.96  129 

9.64  517 

0.35  483 

0.03  871 

0.39  354 

10 

51 

.60  675 

.96  123 

.64  552 

.35  448 

.03  877 

.39  325 

9 

52 

.60  704 

.96  118 

.64  586 

.35  414 

.03  882 

.39  296 

8 

53 

.60  732 

.96  112 

.64  620 

.35  380 

.03  888 

.39  268 

7 

54 

.60  761 

.96  107 

.64  654 

.35  346 

.03  893 

.39  239 

6 

55 

9.60  789 

9.96  101 

9.64  688 

0.35  312 

0.03  899 

0.39  211 

5 

56 

.60  818 

.96  095 

.64  722 

.35  278 

.03  905 

.39  182 

4 

57 

.60  846 

.96  090 

.64  756 

.35  244 

.03  910 

.39  154 

3 

58 

.60  875 

.96  084 

.64  790 

.35  210 

.03  916 

.39  125 

2 

59 

.60  903 

.96  079 

.64  824 

.35  176 

.03  921 

.39  097 

1 

60 

9.60  931 

9.96  073 
Sin 

9.64  858 
Cot 

0.35  142 

0.03  927 
Csc 

0.39  069 

0 

Cos 

Tan 

Sec 

/ 

113°  (293°) 


(246°)  66= 


220 


Table  4.    Trigonometric  Logarithms 


24°  (204°) 

(335°)  155° 

/ 

Sin 

Cos 

Tan 

Cot 

Sec 

Csc 

0 

9.60  931 

9.96  073 

9.64  858 

0.35  142 

0.03  927 

0.39  069 

60 

1 

.60  960 

.96  067 

.64  892 

.35  108 

.03  933 

.39  040 

59 

2 

.60  988 

.96  062 

.64  926 

.35  074 

.03  938 

.39  012 

58 

3 

.61  016 

.96  056 

.64  960 

.35  040 

.03  944 

.38  984 

57 

4 

.61  045 

.96  050 

.64  994 

.35  006 

.03  950 

.38  955 

56 

5 

9.61  073 

9.96  045 

9.65  028 

0.34  972 

0.03  955 

0.38  927 

55 

6 

.61  101 

.96  039 

.65  062 

.34  938 

.03  961 

.38  899 

54 

7 

.61  129 

.96  034 

.65  096 

.34  904 

.03  966 

.38  871 

53 

8 

.61  158 

.96  028 

.65  130 

.34  870 

.03  972 

.38  842 

52 

9 

.61  186 

.96  022 

.65  164 

.34  836 

.03  978 

.38  814 

51 

10 

9.61  214 

9.96  017 

9.65  197 

0.34  803 

0.03  983 

0.38  786 

50 

11 

.61  242 

.96  011 

.65  231 

.34  769 

.03  989 

.38  758 

49 

12 

.61  270 

.96  005 

.65  265 

.34  735 

.03  995 

.38  730 

48 

13 

.61  298 

.96  000 

.65  299 

.34  701 

.04  000 

.38  702 

47 

14 

.61  326 

.95  994 

.65  333 

.34  667 

.04  006 

.38  674 

46 

15 

9.61  354 

9.95  988 

9.65  366 

0.34  634 

0.04  012 

0.38  646 

45 

16 

.61  382 

.95  982 

.65  400 

.34  600 

.04  018 

.38  618 

44 

17 

.61411 

.95  977 

.65  434 

.34  566 

.04  023 

.38  589 

43 

18 

.61  438 

.95  971 

.65  467 

.34  533 

.04  029 

.38  562 

42 

19 

.61  466 

.95  965 

.65  501 

.34  499 

.04  035 

.38  534 

41 

20 

9.61  494 

9.95  960 

9.65  535 

0.34  465 

0.04  040 

0.38  506 

40 

21 

.61  522 

.95  954 

.65  568 

.34  432 

.04  046 

.38  478 

39 

22 

.61  550 

.95  948 

.65  602 

.34  398 

.04  052 

.38  450 

38 

23 

.61  578 

.95  942 

.65  636 

.34  364 

.04  058 

.38  422 

37 

24 

.61  606 

.95  937 

.65  669 

.34  331 

.04  063 

.38  394 

36 

25 

9.61  634 

9.95  931 

9.65  703 

0.34  297 

0.04  069 

0.38  366 

35 

26 

.61  662 

.95  925 

.65  736 

.34  264 

.04  075 

.38  338 

34 

27 

.61  689 

.95  920 

.65  770 

.34  230 

.04  080 

.38  311 

33 

28 

.61  717 

.95  914 

.65  803 

.34  197 

.04  086 

.38  283 

32 

29 

.61  745 

.95  908 

.65  837 

.34  163 

.04  092 

.38  255 

31 

30 

9.61  773 

9.95  902 

9.65  870 

0.34  130 

0.04  098r 

0.38  227 

30 

31 

.61  800 

.95  897 

.65  904 

.34  096 

.04  103 

.38  200 

29 

32 

.61  828 

.95  891 

.65  937 

.34  063 

.04  109 

.38  172 

28 

33 

.61  856 

.95  885 

.65  971 

.34  029 

.04  115 

.38  144 

27 

34 

.61  883 

.95  879 

.66  004 

.33  996 

.04  121 

.38  117 

26 

35 

9.61  911 

9.95  873 

9.66  038 

0..33  962 

0.04  127 

0.38  089 

25 

36 

.61  939 

.95  868 

.66  071 

.33  929 

.04  132 

.38  061 

24 

37 

.61  966 

.95  862 

.66  104 

.33  896 

.04  138 

.38  034 

23 

38 

.61  994 

.95  856 

.66  138 

.33  862 

.04  144 

.38  006 

22 

39 

.62  021 

.95  850 

.66  171 

.33  829 

.04  150 

.37  979 

21 

40 

9.62  049 

9.95  844 

9.66  204 

0.33  796 

0.04  156 

0.37  951 

20 

41 

.62  076 

.95  839 

.66  238 

.33  762 

.04  161 

.37  924 

19 

42 

.62  104 

.95  833 

.66  271 

.33  729 

.04  167 

.37  896 

18 

43 

.62  131 

.95  827 

.66  304 

.33  696 

.04  173 

.37  869 

17 

44 

.62  159 

.95  821 

.66  337 

.33  663 

.04  179 

.37  841 

16 

45 

9.62  186 

9.95  815 

9. .66  371 

0.33  629 

0.04  185 

0.37  814 

15 

46 

.62  214 

.95  810 

.66  404 

.33  596 

.04  190 

.37  786 

14 

47 

.62  241 

.95  804 

.66  437 

.33  563 

.04  196 

.37  759 

13 

48 

.62  268 

.95  798 

.66  470 

.33  530 

.04  202 

.37  732 

12 

49 

.62  296 

.95  792 

.66  503 

.33  497 

.04  208 

.37  704 

11 

50 

9.62  323 

9.95  786 

9.66  537 

0.33  463 

0.04  214 

0.37  677 

10 

51 

.62  350 

.95  780 

.66  570 

.33  430 

.04  220 

.37  650 

9 

52 

.62  377 

.95  775 

.66  603 

.33  397 

.04  225 

.37  623 

8 

53 

.62  405 

.95  769 

.66  636 

.33  364 

.04  231 

.37  595 

7 

54 

.62  432 

.95  763 

.66  669 

.33  331 

.04  237 

.37  568 

6 

55 

9.62  459 

9.95  757 

9.66  702 

0.33  298 

0.04  243 

0.37  541 

5 

56 

.62  486 

.95  751 

.66  735 

.33  265 

.04  249 

.37  514 

4 

57 

.62  513 

.95  745  . 

.66  768 

.33  232 

.04  255 

.37  487 

3 

58 

.62  541 

.95  739 

.66  801 

.33  199 

.04  261 

.37  459 

2 

59 

.62  568 

.95  733 

.66  834 

.33  166 

.04  267 

.37  432 

1 

60 

9.62  595 

9.95  728 

9.66  867 

0.33  133 

0.04  272 

0.37  405 

0 

Cos 

Sin 

Cot 

Tan 

Csc 

Sec 

/ 

114°  (294°) 


(245°)  65= 


Table  4. 


Trigonometric 


Logarithms 


221 


25°  (205°) 

(334°)  154° 

' 

Sin 

Cos 

Tan 

Cot 

Sec 

Csc 

0 

9.62  595 

9.95  728 

9.66  867 

0.33  133 

0.04  272 

0.37  405 

60 

1 

.62  622 

.95  722 

.66  900 

.33  100 

.04  278 

.37  378 

59 

2 

.62  649 

.95  716 

.66  933 

.33  067 

.04  284 

.37  351 

58 

3 

.62  676 

.95  710 

.66  966 

.33  034 

.04  290 

.37  324 

57 

4 

.62  703 

.95  704 

.66  999 

.33  001 

.04  296 

.37  297 

56 

5 

9.62  730 

9.95  698 

9.67  032 

0.32  968 

0.04  302 

0.37  270 

55 

6 

.62  757 

.95  692 

.67  065 

.32  935 

.04  308 

.37  243 

54 

7 

.62  784 

.95  686 

.67  098 

.32  902 

.04  314 

.37  216 

53 

8 

.62  811 

.95  680 

.67  131 

.32  869 

.04  320 

.37  189 

52 

9 

.62  838 

.95  674 

.67  163 

.32  837 

.04  326 

.37  162 

51 

10 

9.62  865 

9.95  668 

9.67  196 

0.32  804 

0.04  332 

0.37  135 

50 

11 

.62  892 

.95  663 

.67  229 

.32  771 

.04  337 

.37  108 

49 

12 

.62  918 

.95  657 

.67  262 

.32  738 

.04  343 

.37  082 

48 

13 

.62  945 

.95  651 

.67  295 

.32  705 

.04  349 

.37  055 

47 

14 

.62  972 

.95  645 

.67  327 

.32  673 

.04  355 

.37  028 

46 

15 

9.62  999 

9.95  639 

9.67  360 

0.32  640 

0.04  361 

0.37  001 

45 

16 

.63  026 

.95  633 

.67  393 

.32  607 

.04  367 

.36  974 

44 

17 

.63  052 

.95  627 

.67  426 

.32  574 

.04  373 

.36  948 

43 

18 

.63  079 

.95  621 

.67  458 

.32  542 

.04  379 

.36  921 

42 

19 

.63  106 

.95  615 

.67  491 

.32  509 

.04  385 

.36  894 

41 

20 

9.63  133 

9.95  609 

9.67  524 

0.32  476 

0.04  391 

0.36  867 

40 

21 

.63  159 

.95  603 

.67  556 

.32  444 

.04  397 

.36  841 

39 

22 

.63  186 

.95  597 

.67  589 

.32  411 

.04  403 

.36  814 

38 

23 

.63  213 

.95  591 

.67  622 

.32  378 

.04  409 

.36  787 

37 

24 

.63  239 

.95  585 

.67  654 

.32  346 

.04  415 

.36  761 

36 

25 

9.63  266 

9.95  579 

9.67  687 

0.32  313 

0.04  421 

0.36  734 

35 

26 

.63  292 

.95  573 

.67  719 

.32  281 

.04  427 

.36  708 

34 

27 

.63  319 

.95  567 

.67  752 

.32  248 

.04  433 

.36  681 

33 

28 

.63  345 

.95  561 

.67  785 

.32  215 

.04  439 

.36  655 

32 

29 

.63  372 

.95  555 

.67  817 

.32  183 

.04  445 

.36  628 

31 

30 

9.63  398 

9.95  549 

9.67  850 

0.32  150 

0.04  451 

0.36  602 

30 

31 

.63  425 

.95  543 

.67  882 

.32  118 

.04  457 

.36  575 

29 

32 

.63  451 

.95  537 

.67  915 

.32  085 

.04  463 

.36  549 

28 

33 

.63  478 

.95  531 

.67  947 

.32  053 

.04  469 

.36  522 

27 

34 

.63  504 

.95  525 

.67  980 

.32  020 

.04  475 

.36  496 

26 

35 

9.63  531 

9.95  519 

9.68  012 

0.31  988 

0.04  481 

0.36  469 

25 

36 

.63  557 

.95  513 

.68  044 

.31  956 

.04  487 

.36  443 

24 

37 

.63  583 

.95  507 

.68  077 

.31  923 

.04  493 

.36  417 

23 

38 

.63  610 

.95  500 

.68  109 

.31  891 

.04  500 

.36  390 

22 

39 

.63  636 

.95  494 

.68  142 

.31  858 

.04  506 

.36  364 

21 

40 

9.63  662 

9.95  488 

9.68  174 

0.31  826 

0.04  512 

0.36  338 

20 

41 

.63  689 

.95  482 

.68  206 

.31  794 

.04  518 

.  36  311 

19 

42 

.63  715 

.95  476 

.68  239 

.31  761 

.04  524 

.36  285 

18 

43 

.63  741 

.95  470 

.68  271 

.31  729 

.04  530 

.36  259 

17 

44 

.63  767 

.95  464 

.68  303 

.31  697 

.04  536 

.36  233 

16 

45 

9.63  794 

9.95  458 

9.68  336 

0.31  664 

0.04  542 

0.36  206 

15 

46 

.63  820 

.95  452 

.68  368 

.31  632 

.04  548 

.36  180 

14 

47 

.63  846 

.95  446 

.68  400 

.31  600 

.04  554 

.36  154 

13 

48 

.63  872 

.95  440 

.68  432 

.31  568 

.04  560 

.36  128 

12 

49 

.63  898 

.95  434 

.68  465 

.31  535 

.04  566 

.36  102 

11 

50 

9.63  924 

9.95  427 

9.68  497 

0.31  503 

0.04  573 

0.36  076 

10 

51 

.63  950 

.95  421 

.68  529 

.31  471 

.04  579 

.36  050 

9 

52 

.63  976 

.95  415 

.68  561 

.31  439 

.04  585 

.36  024 

8 

53 

.64  002 

.95  409 

.68  593 

.31  407 

.04  591 

.35  998 

7 

54 

.64  028 

.95  403 

.68  626 

.31  374 

.04  597 

.35  972 

6 

55 

9.64  054 

9.95  397 

9.68  658 

0.31  342 

0.04  603 

0.35  946 

5 

56 

.64  080 

.95  391 

.68  690 

.31  310 

.04  609 

.35  920 

4 

57 

.64  106 

.95  384 

.68  722 

.31  278 

.04  616 

.35  894 

3 

58 

.64  132 

.95  378 

.68  754 

.31  246 

.04  622 

.35  868 

2 

59 

.64  158 

.95  372 

.68  786 

.31  214 

.04  628 

.35  842 

1 

60 

9.64  184 

9.95  366 

9.68  818 

0.31  182 

0.04  634 

0.35  816 

0 

Cos 

Sin 

Cot 

Tan 

Csc 

Sec 

/ 

115°  (295°) 


(244°)  64^ 


222 


flT 


Table  4.    Trigonometric  Logarithms 


26°  (206°) 

(333°)  153° 

/ 

Sin 

Cos 

Tan 

Cot 

Sec 

Csc 

0 

9.64  184 

9.95  366 

9.68  818 

0.31  182 

0.04  634 

0.35  816 

60 

1 

.64  210 

.95  360 

.68  850 

.31  150 

.04  640 

.35  790 

59 

2 

.64  236 

.95  354 

.68  882 

.31  118 

.04  646 

.35  764 

58 

3 

.64  262 

.95  348 

.68  914 

.31  086 

.04  652 

.35  738 

57 

4 

.64  288 

.95  341 

.68  946 

.31  054 

.04  659 

.35  712 

56 

5 

9.64  313 

9.95  335 

9.68  978 

0.31  022 

0.04  665 

0.35  687 

55 

6 

.64  339 

.95  329 

.69  010 

.30  990 

.04  671 

.35  661 

54 

7 

.64  365 

.95  323 

.69  042 

.30  958 

.04  677 

.35  635 

53 

8 

.64  391 

.95  317 

.69  074 

.30  926 

.04  683 

.35  609 

52 

9 

.64  417 

.95  310 

.69  106 

.30  894 

.04  690 

.35  583 

51 

10 

9.64  442 

9.95  304 

9.69  138 

0.30  862 

0.04  696 

0.35  558 

50 

11 

.64  468 

.95  298 

.69  170 

.30  830 

.04  702 

.35  532 

49 

12 

.64  494 

.95  292 

.69  202 

.30  798 

.04  708 

.35  506 

48 

13 

.64  519 

.95  286 

.69  234 

.30  766 

.04  714 

.35  481 

47 

14 

.64  545 

.95  279 

.69  266 

.30  734 

.04  721 

.35  455 

46 

15 

9.64  571 

9.95  273 

9.69  298 

0.30  702 

0.04  727 

0.35  429 

45 

16 

.64  596 

.95  267 

.69  329 

.30  671 

.04  733 

.35  404 

44 

17 

.64  622 

.95  261 

.69  361 

.30  639 

.04  739 

.35  378 

43 

18 

.64  647 

.95  254 

.69  393 

.30  607 

.04  746 

.35  353 

42 

19 

.64  673 

.95  248 

.69  425 

.30  575 

.04  752 

.35  327 

41 

20 

9.64  698 

9.95  242 

9.69  457 

0.30  543 

0.04  758 

0.35  302 

40 

21 

.64  724 

.95  236 

.69  488 

.30  512 

.04  764 

.35  276 

39 

22 

.64  749 

.95  229 

.69  520 

.30  480 

.04  771 

.35  251 

38 

23 

.64  775 

.95  223 

.69  552 

.30  448 

.04  777 

.35  225 

37 

24 

.64  800 

.95  217 

.69  584 

.30  416 

.04  783 

.35  200 

36 

25 

9.64  826 

9.95  211 

9.69  615 

0,30  385 

0.04  789 

0.35  174 

35 

26 

.64  851 

.95  204 

.69  647 

.30  353 

.04  796 

.35  149 

34 

27 

.64  877 

.95  198 

.69  679 

.30  321 

.04  802 

.35  123 

33 

28 

.64  902 

.95  192 

.69  710 

.30  290 

.04  808 

.35  098 

32 

29 

.64  927 

.95  185 

.69  742 

.30  258 

.04  815 

.35  073 

31 

30 

9.64  953 

9.95  179 

9.69  774 

0.30  226 

0.04  821 

0.35  047 

30 

31 

.64  978 

.95  173 

.69  805 

.30  195 

.04  827 

.35  022 

29 

32 

.65  003 

.95  167 

.69  837 

.30  163 

.04  833 

.34  997 

28 

33 

.65  029 

.95  160 

.69  868 

.30  132 

.04  840 

.34  971 

27 

34 

.65  054 

.95  154 

.69  900 

.30  100 

.04  846 

.34  946 

26 

35 

9.65  079 

9.95  148 

9.69  932 

0.30  068 

0.04  852 

0.34  921 

25 

36 

.65  104 

.95  141 

.69  963 

.30  037 

.04  859 

.34  896 

24 

37 

.65  130 

.95  135 

.69  995 

.30  005 

.04  865 

.34  870 

23 

38 

.65  155 

.95  129 

.70  026 

.29  974 

.04  871 

.34  845 

22 

39 

.65  180 

.95  122 

.70  058 

.29  942 

.04  878 

.34  820 

21 

40 

9.65  205 

9.95  116 

9.70  089 

0.29  911 

0.04  884 

0.34  795 

20 

41 

.65  230 

.95  110 

.70  121 

.29  879 

.04  890 

.34  770 

19 

42 

.65  255 

.95  103 

.70  152 

.29  848 

.04  897 

.34  745 

18 

43 

.65  281 

.95  097 

.70  184 

.29  816 

.04  903 

.34  719 

17 

44 

.65  306 

.95  090 

.70  215 

.29  785 

.04  910 

.34  694 

16 

45 

9.65  331 

9.95  084 

9.70  247 

0.29  753 

0.04  916 

0.34  669 

15 

46 

.65  356 

.95  078 

.70  278 

.29  722 

.04  922 

.34  644 

14 

47 

.65  381 

.95  071 

.70  309 

.29  691 

.04  929 

.34  619 

13 

48 

.65  406 

.95  065 

.70  341 

.29  659 

.04  935 

.34  594 

12 

49 

.65  431 

.95  059 

.70  372 

.29  628 

.04  941 

.34  569 

11 

50 

9.65  456 

9.95  052 

9.70  404 

0.29  596 

0.04*948 

0.34  544 

10 

51 

.65  481 

.95  046 

.70  435 

.29  565 

.04  954 

.34  519 

9 

52 

.65  506 

.95  039 

.70  466 

.29  534 

.04  961 

.34  494 

8 

53 

.65  531 

.95  033 

.70  498 

.29  502 

.04  967 

.34  469 

7 

54 

.65  556 

.95  027 

.70  529 

.29  471 

.04  973 

.34  444 

6 

55 

9.65  580 

9.95  020 

9.70  560 

0.29  440 

0.04  980 

0.34  420 

5 

56 

.65  605 

.95  014 

.70  592 

.29  408 

.04  986 

.34  395 

4 

57 

.65  630 

.95  007 

.70  623 

.29  377 

.04  993 

.34  370 

3 

58 

.65  655 

.95  001 

.70  654 

.29  346 

.04  999 

.34  345 

2 

59 

.65  680 

.94  995 

.70  685 

.29  315 

.05  005 

.34  320 

1 

60 

9.65  705 

9.94  988 

9.70  717 

0.29  283 

0.05  012 

0.34  295 

0 

Cos 

Sin 

Cot 

Tan 

Csc 

Sec 

/ 

116°  (296°) 


(243°)  63° 


Table  4.    Trigonometric  Logarithms 


223 


27°  (207°) 

(332°)  152° 

/ 

Sin 

Cos 

Tan 

Cot 

Sec 

Csc 

0 

9.65  705 

9.94  988 

9.70  717 

0.29  283 

0.05  012 

0.34  295 

60 

1 

.65  729 

.94  982 

.70  748 

.29  252 

.05  018 

.34  271 

59 

2 

.65  754 

.94  975 

.70  779 

.29  221 

.05  025 

.34  246 

58 

3 

.65  779 

.94  969 

.70  810 

.29  190 

.05  031 

.34  221 

57 

4 

.65  804 

.94  962 

.70  841 

.29  159 

.05  038 

.34  196 

56 

5 

9.65  828 

9.94  956 

9.70  873 

0.29  127 

0.05  044 

0.34  172 

55 

6 

.65  853 

.94  949 

.70  904 

.29  090 

.05  051 

.34  147 

54 

7 

.65  878 

.94  943 

.70  935 

.29  065 

.05  057 

.34  122 

53 

8 

.65  902 

.94  936 

.70  966 

.29  034 

.05  064 

.34  098 

52 

9 

.65  927 

.94  930 

.70  997 

.29  003 

.05  070 

.34  073 

51 

10 

9.65  952 

9.94  923 

9.71  028 

0.28  972 

0.05  077 

0.34  048 

50 

11 

.65  976 

.94  917 

.71  059 

.28  941 

.05  083 

.34  024 

49 

12 

.66  001 

.94  911 

.71  090 

.28  910 

.05  089 

.33  999 

48 

13 

.66  025 

.94  904 

.71  121 

.28  879 

.05  096 

.33  975 

47 

14 

.66  050 

.94  898 

.71  153 

.28  847 

.05  102 

.33  950 

46 

15 

9.66  075 

9.94  891 

9.71  184 

0.28  816 

0.05  109 

0.33  925 

45 

16 

.66  099 

.94  885 

.71  215 

.28  785 

.05  115 

.33  901 

44 

17 

.66  124 

.94  878 

.71  246 

.28  754 

.05  122 

.33  876 

43 

18 

.66  148 

.94  871 

.71  277 

.28  723 

.05  129 

.33  852 

42 

19 

.66  173 

.94  865 

.71  308 

.28  692 

.05  135 

.33  827 

41 

20 

9.66  197 

9.94  858 

9.71  339 

0.28  661 

0.05  142 

0.33  803 

40 

21 

.66  221 

.94  852 

.71  370 

.28  630 

.05  148 

.33  779 

39 

22 

.66  246 

.94  845 

.71  401 

.28  599 

.05  155 

.33  754 

38 

23 

.66  270 

.94  839 

.71  431 

.28  569 

.05  161 

.33  730 

37 

24 

.66  295 

.94  832 

.71  462 

.28  538 

.05  168 

.33  705 

36 

25 

9.66  319 

9.94  826 

9.71  493 

0.28  507 

0.05  174 

0.33  681 

35 

26 

.66  343 

.94  819 

.71  524 

.28  476 

.05  181 

.33  657 

34 

27 

.66  368 

.94  813 

.71  555 

.28  445 

.05  187 

.33  632 

33 

28 

.66  392 

.94  806 

.71  586 

.28  414 

.05  194 

.33  608 

32 

29 

.66  416 

.94  799 

.71  617 

.28  383 

.05  201 

.33  584 

31 

30 

9.66  441 

9.94  793 

9.71  648 

0.28  352 

0.05  207 

0.33  559 

30 

31 

.66  465 

.94  786 

.71  679 

.28  321 

.05  214 

.33  535 

29 

32 

.66  489 

.94  780 

.71  709 

.28  291 

.05  220 

.33  511 

28 

33 

.66  513 

.94  773 

.71  740 

.28  260 

.05  227 

.33  487 

27 

34 

.66  537 

.94  767 

.71  771 

.28  229 

.05  233 

.33  463 

26 

35 

9.66  562 

9.94  760 

9.71  802 

0.28  198 

0.05  240 

0.33  438 

25 

36 

.66  586 

.94  753 

.71  833 

.28  167 

.05  247 

.33  414 

24 

37 

.66  610 

.94  747 

.71  863 

.28  137 

.05  253 

.33  390 

23 

38 

.66  634 

.94  740 

.71  894 

.28  106 

.05  260 

.33  366 

22 

39 

.66  658 

.94  734 

.71  925 

.28  075 

.05  266 

.33  342 

21 

40 

9.66  682 

9.94  727 

9.71  955 

0.28  045 

0.05  273 

0.33  318 

20 

41 

.66  706 

.94  720 

.71  986 

.28  014 

.05  280 

.33  294 

19 

42 

.66  731 

.94  714 

.72  017 

.27  983 

.05  286 

.33  269 

18 

43 

.66  755 

.94  707 

.72  048 

.27  952 

.05  293 

.33  245 

17 

44 

.66  779 

.94  700 

.72  078 

.27  922 

.05  300 

.33  221 

16 

45 

9.66  803 

9.94  694 

9.72  109 

0.27  891 

0.05  306 

0.33  197 

15 

46 

.66  827 

.94  687 

.72  140 

.27  860 

.05  313 

.33  173 

14 

47 

.66  851 

.94  680 

.72  170 

.27  830 

.05  320 

.33  149 

13 

48 

.66  875 

.94  674 

.72  201 

.27  799 

.05  326 

.33  125 

12 

49 

.66  899 

.94  667 

.72  231 

.27  769 

.05  333 

.33  101 

11 

50 

9.66  922 

9.94  660 

9.72  262 

0.27  738 

0.05  340 

0.33  078 

10 

51 

.66  946 

.94  654 

.72  293 

.27  707 

.05  346 

.33  054 

9 

52 

.66  970 

.94  647 

.72  323 

.27  677 

.05  353 

.33  030 

8 

53 

.66  994 

.94  640 

.72  354 

.27  646 

.05  360 

.33  006 

7 

54 

.67  018 

.94  634 

.72  384 

.27  616 

.05  366 

.32  982 

6 

55 

9.67  042 

9.94  627 

9.72  415 

0.27  585 

0.05  373 

0.32  958 

5 

56 

.67  066 

.94  620 

.72  445 

.27  555 

.05  380 

.32  934 

4 

57 

.67  090 

.94  614 

.72  476 

.27  524 

.05  386 

.32  910 

3 

58 

.67  113 

.94  607 

.72  506 

.27  494 

.05  393 

.32  887 

2 

59 

.67  137 

.94  600 

.72  537 

.27  463 

.05  400 

.32  863 

1 

60 

9.67  161 

9.94  593 

9.72  567 

0.27  433 

0.05  407 

0.32  839 

0 

Cos 

Sin 

Cot 

Tan 

Csc 

Sec 

' 

117°  (297°) 


(242')  62° 


224 


Table  4.    Trigonometric  Losrarithms 


28°  (208°) 


(331°)  151° 


' 

Sin 

Cos 

Tan 

Cot 

Sec 

Csc 

0 

9.67  161 

9.94  593 

9.72  567 

0.27  433 

0.05  407 

0.32  839 

60 

1 

.67  185 

.94  587 

.72  598 

.27  402 

.05  413 

.32  815 

59 

2 

.67  208 

.94  580 

.72  628 

.27  372 

.05  420 

.32  792 

58 

3 

.67  232 

.94  573 

.72  659 

.27  341 

.05  427 

.32  768 

57 

4 

.67  256 

.94  567 

.72  689 

.27  311 

.05  433 

.32  744 

56 

5 

9.67  280 

9.94  560 

9.72  720 

0.27  280 

0.05  440 

0.32  720 

55 

6 

.67  303 

.94  553 

.72  750 

.27  250 

.05  447 

.32  697 

54 

7 

.67  327 

.94  546 

.72  780 

.27  220 

.05  454 

.32  673 

53 

8 

.67  350 

.94  540 

.72  811 

.27  189 

.05  460 

.32  650 

52 

9 

.67  374 

.94  533 

.72  841 

.27  159 

.05  467 

.32  626 

51 

10 

9.67  398 

9.94  526 

9.72  872 

0.27  128 

0.05  474 

0.32  602 

50 

11 

.67  421 

.94  519 

.72  902 

.27  098 

.05  481 

.32  579 

49 

12 

.67  445 

.94  513 

.72  932 

.27  068 

.05  487 

.32  555 

48 

13 

.67  468 

.94  506 

.72  963 

.27  037 

.05  494 

.32  532 

47 

14 

.67  492 

.94  499 

.72  993 

.27  007 

.05  501 

.32  508 

46 

15 

9.67  515 

9.94  492 

9.73  023 

0.26  977 

0.05  508 

0.32  485 

45 

16 

.67  539 

.94  485 

.73  054 

.26  946 

.05  515 

.32  461 

44 

17 

.67  562 

.94  479 

.73  084 

.26  916 

.05  521 

.32  438 

43 

18 

.67  586 

.94  472 

.73  114 

.26  886 

.05  528 

.32  414 

42 

19 

.67  609 

.94  465 

.73  144 

.26  856 

.05  535 

.32  391 

41 

20 

9.67  633 

9.94  458 

9.73  175 

0.26  825 

0.05  542 

0.32  367 

40 

21 

.67  656 

.94  451 

.73  205 

.26  795 

.05  549 

.32  344 

39 

22 

.67  680 

.94  445 

.73  235 

.26  765 

.05  555 

.32  320 

38 

23 

.67  703 

.94  438 

.73  265 

.26  735 

.05  562 

.32  297 

37 

24 

.67  726 

.94  431 

.73  295 

.26  705 

.05  569 

.32  274 

36 

25 

9.67  750 

9.94  424 

9.73  326 

0.26  674 

0.05  576 

0.32  250 

35 

26 

.67  773 

.94  417 

.73  356 

.26  644 

.05  583 

.32  227 

34 

27 

.67  796 

.94  410 

.73  386 

.26  614 

.05  590 

.32  204 

33 

28 

.67  820 

.94  404 

.73  416 

.26  584 

.05  596 

.32  180 

32 

29 

.67  843 

.94  397 

.73  446 

.26  554 

.05  603 

.32  157 

31 

30 

9.67  866 

9.94  390 

9.73  476 

0.26  524 

0.05  610 

0.32  134 

30 

31 

.67  890 

.94  383 

.73  507 

.26  493 

.05  617 

.32  110 

29 

32 

.67  913 

.94  376 

.73  537 

.26  463 

.05  624 

.32  087 

28 

33 

.67  936 

.94  369 

.73  567 

.26  433 

.05  631 

.32  064 

27 

34 

.67  959 

.94  362 

.73  597 

.26  403 

.05  638 

.32  041 

26 

35 

9.67  982 

9.94  355 

9.73  627 

0.26  373 

0.05  645 

0.32  018 

25 

36 

.68  006 

.94  349 

.73  657 

.26  343 

.05  651 

.31  994 

24 

37 

.68  029 

.94  342 

.73  687 

.26  313 

.05  658 

.31  971 

23 

38 

.68  052 

.94  335 

.73  717 

.26  283 

.05  665 

.31  948 

22 

39 

.68  075 

.94  328 

.73  747 

.26  253 

.05  672 

.31  925 

21 

40 

9.68  098 

9.94  321 

9.73  777 

0.26  223 

0.05  679 

0.31  902 

20 

41 

.68  121 

.94  314 

.73  807 

.26  193 

.05  686 

.31  879 

19 

42 

.68  144 

.94  307 

.73  837 

.26  163 

.05  693 

.31  856 

18 

43 

.68  167 

.94  300 

.73  867 

.26  133 

.05  700 

.31  833 

17 

44 

.68  190 

.94  293 

.73  897 

.26  103 

.05  707 

.31  810 

16 

45 

9.68  213 

9.94  286 

9.73  927 

0.26  073 

0.05  714 

0.31  787 

15 

46 

.68  237 

.94  279 

.73  957 

.26  043 

.05  721 

.31  763 

14 

47 

.68  260 

.94  273 

.73  987 

.26  013 

.05  727 

.31  740 

13 

48 

.68  283 

.94  266 

.74  017 

.25  983 

.05  734 

.31  717 

12 

49 

.68  305 

.94  259 

.74  047 

.25  953 

.05  741 

.31  695 

11 

50 

9.68  328 

9.94  252 

9.74  077 

0.25  923 

0.05  748 

0.31  672 

10 

51 

.68  351 

.94  245 

.74  107 

.25  893 

.05  755 

.31  649 

9 

52 

.68  374 

.94  238 

.74  137 

.25  863 

.05  762 

.31  626 

8 

53 

.68  397 

.94  231 

.74  166 

.25  834 

.05  769 

.31  603 

7 

54 

.68  420 

.94  224 

.74  196 

.25  804 

.05  776 

.31  580 

6 

55 

9.68  443 

9.94  217 

9.74  226 

0.25  774 

0.05  783 

0.31  557 

5 

56 

.68  466 

.94  210 

.74  256 

.25  744 

.05  790 

.31  534 

4 

57 

.68  489 

.94  203 

.74  286 

.25  714 

.05  797 

.31  511 

3 

58 

.68  512 

.94  196 

.74  316 

.25  684 

.05  804 

.31  488 

2 

59 

.68  534 

.94  189 

.74  345 

.25  655 

.05  811 

.31  466 

1 

60 

9.68  557 

9.94  182 

9.74  375 

0.25  625 

0.05  818 

0.31  443 

0 

Cos 

Sin 

Cot 

Tan 

Csc 

Sec 

' 

118°  (298°) 


(241°)  61' 


Table  4. 


Trigonometric  Logarithms 


225 


29°  (209°) 

(330°)  150° 

/ 

Sin 

Cos 

Tan 

Cot 

Sec 

Csc 

0 

9.68  557 

9.94  182 

9.74  375 

0.25  625 

0.05  818 

0.31  443 

60 

1 

.68  580 

.94  175 

.74  405 

.25  595 

.05  825 

.31  420 

59 

2 

.68  603 

.94  168 

.74  435 

.25  565 

.05  832 

.31  397 

58 

3 

.68  625 

.94  161 

.74  465 

.25  535 

.05  839 

.31  375 

57 

4 

.68  648 

.94  154 

.74  494 

.25  506 

.05  846 

.31  352 

56 

5 

9.68  671 

9.94  147 

9.74  524 

0.25  476 

0.05  853 

0.31  329 

55 

6 

.68  694 

.94  140 

.74  554 

.25  446 

.05  860 

.31  306 

54 

7 

.68  716 

.94  133 

.74  583 

.25  417 

.05  867 

.31  284 

53 

8 

.68  739 

.94  126 

.74  613 

.25  387 

.05  874 

.31  261 

52 

9 

.68  762 

.94  119 

.74  643 

.25  357 

.05  881 

.31  238 

51 

10 

9.68  784 

9.94  112 

9.74  673 

0.25  327 

0.05  888 

0.31  216 

50 

11 

.68  807 

.94  105 

.74  702 

.25  298 

.05  895 

.31  193 

49 

12 

.68  829 

.94  098 

.74  732 

.25  268 

.05  902 

.31  171 

48 

13 

.68  852 

.94  090 

.74  762 

.25  238 

.05  910 

.31  148 

47 

14 

.68  875 

.94  083 

.74  791 

.25  209 

.05  917 

.31  125 

46 

15 

9.68  897 

9.94  076 

9.74  821 

0.25  179 

0.05  924 

0.31  103 

45 

16 

.68  920 

.94  069 

.74  851 

.25  149 

.05  931 

-.31  080 

44 

17 

.68  942 

.94  062 

.74  880 

.25  120 

.05  938 

.31  058 

43 

18 

.68  965 

.94  055 

.74  910 

.25  090 

.05  945 

.31  035 

42 

19 

.68  987 

.94  048 

.74  939 

.25  061 

.05  952 

.31  013 

41 

20 

9.69  010 

9.94  041 

9.74  969 

0.25  031 

0.05  959 

0.30  990 

40 

21 

.69  032 

.94  034 

.74  998 

.25  002 

.05  966 

.30  968 

39 

22 

.69  055 

.94  027 

.75  028 

.24  972 

.05  973 

.30  945 

38 

23 

.69  077 

.94  020 

.75  058 

.24  942 

.05  980 

.30  923 

37 

24 

.69  100 

.94  012 

.75  087 

.24  913 

.05  988 

.30  900 

36 

25 

9.69  122 

9.94  005 

9.75  117 

0.24  883 

0.05  995 

0.30  878 

35 

26 

.69  144 

.93  998 

.75  146 

.24  854 

.06  002 

.30  856 

34 

27 

.69  167 

.93  991 

.75  176 

.24  824 

.06  009 

.30  833 

33 

28 

.69  189 

.93  984 

.75  205 

.24  795 

.06  016 

.30  811 

32 

29 

.69  212 

.93  977 

.75  235 

.24  765 

.06  023 

.30  788 

31 

30 

9.69  234 

9.93  970 

9.75  264 

0.24  736 

0.06  030 

0.30  766 

30 

31 

.69  256 

.93  963 

.75  294 

.24  706 

.06  037 

.30  744 

29 

32 

.69  279 

.93  955 

.75  323 

.24  677 

.06  045 

.30  721 

28 

33 

.69  301 

.93  948 

.75  353 

.24  647 

.06  052 

.30  699 

27 

34 

.69  323 

.93  941 

.75  382 

.24  618 

.06  059 

.30  677 

26 

35 

9.69  345 

9.93  934 

9.75  411 

0.24  589 

0.06  066 

0.30  055 

25 

36 

.69  368 

.93  927 

.75  441 

.24  559 

.06  073 

.30  632 

24 

37 

.69  390 

.93  920 

.75  470 

.24  530 

.06  080 

.30  610 

23 

38 

.69  412 

.93  912 

.75  500 

.24  500 

.06  088 

.30  588 

22 

39 

.69  434 

.93  905 

.75  529 

.24  471 

.06  095 

.30  566 

21 

40 

9.69  456 

9.93  898 

9.75  558 

0.24  442 

0.06  102 

0.30  544 

20 

41 

.69  479 

.93  891 

.75  588 

.24  412 

.06  109 

.30  521 

19 

42 

.69  501 

.93  884 

.75  617 

.24  383 

.06  116 

.30  499 

18 

43 

.69  523 

.93  870 

.75  647 

.24  353 

.06  124 

.30  477 

17 

44 

.69  545 

.93  869 

.75  676 

.24  324 

.06  131 

.30  455 

16 

45 

9.69  567 

9.93  862 

9.75  705 

0.24  295 

0.06  138 

0.30  433 

15 

46 

.69  589 

.93  855 

.75  735 

.24  265 

.06  145 

.30  411 

14 

47 

.69  611 

.93  847 

.75  764 

.24  236 

.06  153 

.30  389 

13 

48 

.69  633 

.93  840 

.75  793 

.24  207 

.06  160 

.30  367 

12 

49 

.69  655 

.93  833 

.75  822 

.24  178 

.06  167 

.30  345 

11 

50 

9.69  077 

9.93  826 

9.75  852 

0.24  148 

0.06  174 

0.30  323 

10 

51 

.69  699 

.93  819 

.75  881 

.24  119 

.06  181 

.30  301 

9 

52 

.69  721 

.93  811 

.75  910 

.24  090 

.06  189 

.30  279 

8 

53 

.69  743 

.93  804 

.75  939 

.24  061 

.06  196 

.30  257 

7 

54 

.69  765 

.93  797 

.75  969 

.24  031 

.06  203 

.30  235 

6 

55 

9.69  787 

9.93  789 

9.75  998 

0.24  002 

0.06  211 

0.30  213 

5 

56 

.69  809 

.93  782 

.76  027 

.23  973 

.06  218 

.30  191 

4 

57 

.69  831 

.93  775 

.76  056 

.23  944 

.06  225 

.30  169 

3 

58 

.69  853 

.93  768 

.76  086 

.23  914 

.06  232 

.30  147 

2 

59 

.69  875 

.93  760 

.76  115 

.23  885 

.06  240 

.30  125 

1 

60 

9.69  897 

9.93  753 

0.76  144 

0.23  856 

0.06  247 

0.30  103 

0 

Cos 

Sin    f 

Cot 

Tan 

Csc 

Sec 

1 

119°  (299°) 


(240°)  60' 


226 


Table  4.    Trigouometric  Logarithms 


30°  (210°) 

' 

(329°)  149° 

/ 

Sin 

Cos 

Tan 

Cot 

Sec 

Csc 

0 

9.69  897 

9.93  753 

9.76  144 

0.23  856 

0.06  247 

0.30  103 

60 

1 

.69  919 

.93  746 

.76  173 

.23  827 

.06  254 

.30  081 

59 

2 

.69  941 

.93  738 

.76  202 

.23  798 

.06  262 

.30  059 

58 

3 

.69  963 

.93  731 

.76  231 

.23  769 

.06  269 

.30  037 

57 

4 

.69  984 

.93  724 

.76  261 

.23  739 

.06  276 

.30  016 

56 

5 

9.70  006 

9.93  717 

9.76  290 

0.23  710 

0.06  283 

0.29  994 

55 

6 

.70  028 

.93  709 

.76  319 

.23  681 

.06  291 

.29  972 

54 

7 

.70  050 

.93  702 

.76  348 

.23  652 

.06  298 

.29  950 

53 

8 

.70  072 

.93  695 

.76  377 

.23  623 

.06  305 

.29  928 

52 

9 

.70  093 

.93  687 

.76  406 

.23  594 

.06  313 

.29  907 

51 

10 

9.70  115 

9.93  680 

9.76  435 

0.23  565 

0.06  320 

0.29  885 

50 

11 

.70  137 

.93  673 

.76  464 

.23  536 

.06  327 

.29  863 

49 

12 

.70  159 

.93  665 

.76  493 

.23  507 

.06  335 

.29  841 

48 

13 

.70  180 

.93  658 

.76  522 

.23  478 

.06  342 

.29  820 

45 

14 

.70  202 

.93  650 

.76  551 

.23  449 

.06  350 

.29  798 

46 

15 

9.70  224 

9.93  643 

9.76  580 

0.23  420 

0.06  357 

0.29  776 

45 

16 

.70  245 

.93  636 

.76  609 

.23  391 

.06  364 

.29  755 

44 

17 

.70  267 

.93  628 

.76  639 

.23  361 

.06  372 

.29  733 

43 

18 

.70  288 

.93  621 

.76  668 

.23  332 

.06  379 

.29  712 

42 

19 

.70  310 

.93  614 

.76  697 

.23  303 

.06  386 

.29  690 

41 

20 

9.70  332 

9.93  606 

9.76  725 

0.23  275 

0.06  394 

0.29  668 

40 

21 

.70  353 

.93  599 

.76  754 

.23  246 

.06  401 

.29  647 

39 

22 

.70  375 

.93  591 

.76  783 

.23  217 

.06  409 

.29  625 

38 

23 

.70  396 

.93  584 

.76  812 

.23  188 

.06  416 

.29  604 

37 

24 

.70  418 

.93  577 

.76  841 

.23  159 

.06  423 

.29  582 

36 

25 

9.70  439 

9.93  569 

9.76  870 

0.23  130 

0.06  431 

0.29  561 

35 

26 

.70  461 

.93  562 

.76  899 

.23  101 

.06  438 

.29  539 

34 

27 

.70  482 

.93  554 

.76  928 

.23  072 

.06  446 

.29  518 

33 

28 

.70  504 

.93  547 

.76  957 

.23  043 

.06  453 

.29  496 

32 

29 

.70  525 

.93  539 

.76  986 

.23  014 

.06  461 

.29  475 

31 

30 

9.70  547 

9.93  532 

9.77  015 

0.22  985 

0.06  468 

0.29  453 

30 

31 

.70  568 

.93  525 

.77  044 

.22  956 

.06  475 

.29  432 

29 

32 

.70  590 

.93  517 

.77  073 

.22  927 

.06  483 

.29  410 

28 

33 

.70  611 

.93  510 

.77  101 

.22  899 

.06  490 

.29  389 

27 

34 

.70  633 

.93  502 

.77  130 

.22  870 

.06  498 

.29  367 

26 

35 

9.70  654 

9.93  495 

9.77  159 

0.22  841 

0.06  505 

0.29  346 

25 

36 

.70  675 

.93  487 

.77  188 

.22  812 

.06  513 

.29  325 

24 

37 

.70  697 

.93  480 

.77  217 

.22  783 

.06  520 

.29  303 

23 

38 

.70  718 

.93  472 

.77  246 

.22  754 

.06  528 

.29  282 

22 

39 

.70  739 

.93  465 

.77  274 

.22  726 

.06  535 

.29  261 

21 

40 

9.70  761 

9.93  457 

9.77  303 

0.22  697 

0.06  543 

0.29  239 

20 

41 

.70  782 

.93  450 

.77  332 

.22  668 

.06  550 

.29  218 

19 

42 

.70  803 

.93  442 

.77  361 

.22  639 

.06  558 

.29  197 

18 

43 

.70  824 

.93  435 

.77  390 

.22  610 

.06  565 

.29  176 

17 

44 

.70  846 

.93  427 

.77  418 

.22  582 

.06  573 

.29  154 

16 

45 

9.70  867 

9.93  420 

9.77  447 

0.22  553 

0.06  580 

0.29  133 

15 

46 

.70  888 

.93  412 

.77  476 

.22  524 

.06  588 

.29  112 

14 

47 

.70  909 

.93  405 

.77  505 

.22  495 

.06  595 

.29  091 

13 

48 

.70  931 

.93  397 

.77  533 

.22  467 

.06  603 

.29  069 

12 

49 

.70  952 

.93  390 

.77  562 

.22  438 

.06  610 

.29  048 

11 

50 

9.70  973 

9.93  382 

9.77  591 

0.22  409 

0.06  618 

0.29  027 

10 

51 

.70  994 

.93  375 

.77  619 

.22  381 

.06  625 

.29  006 

9 

52 

.71  015 

.93  367 

.77  648 

.22  352 

.06  633 

.28  985 

8 

53 

.71  036 

.93  360 

.77  677 

.22  323 

.06  640 

.28  964 

7 

54 

.71  058 

.93  352 

.77  706 

.22  294 

.06  648 

.28  942 

6 

55 

9.71  079 

9.93  344 

9.77  734 

0.22  266 

0.06  656 

0.28  921 

5 

56 

.71  100 

.93  337 

.77  763 

.22  237 

.06  663 

.28  900 

4 

57 

.71  121 

.93  329 

.77  791 

.22  209 

.06  671 

.28  879 

3 

58 

.71  142 

.93  322 

.77  820 

.22  180 

.06  678 

.28  858 

2 

59 

.71  163 

.93  314 

.77  849 

.22  151 

.06  686 

.28  837 

1 

60 

9.71  184 

9.93  307 

9.77  877 
Cot 

0.22  123 

0.06  693 

0.28  816 
Sec 

0 

Cos 

Sin 

Tan 

Csc 

/ 

120°  (300°) 


(239°)  59"^ 


Table  4.    Trigonometric  Logarithms 


227 


31°  (211°) 


(328°)  148° 


/ 

Sin 

Cos 

Tan 

Cot 

Sec 

Csc 

0 

9.71  184 

9.93  307 

9.77  877 

0.22  123 

0.06  693 

0.28  816 

60 

1 

.71  205 

.93  299 

.77  906 

.22  094 

.06  701 

.28  795 

59 

2 

.71  226 

.93  291 

.77  935 

.22  065 

.06  709 

.28  774 

58 

3 

.71  247 

.93  284 

.77  963 

.22  037 

.06  716 

.28  753 

57 

4 

.71  268 

.93  276 

.77  992 

.22  008 

.06  724 

.28  732 

56 

5 

9.71  289 

9.93  269 

9.78  020 

0.21  980 

0.06  731 

0.28  711 

55 

6 

.71  310 

.93  261 

.78  049 

.21  951 

.06  739 

.28  690 

54 

7 

.71  331 

.93  253 

.78  077 

.21  923 

.06  747 

.28  669 

53 

8 

.71  352 

.93  246 

.78  106 

.21  894 

.06  754 

.28  648 

52 

9 

.71  373 

.93  238 

.78  135 

.21  865 

.06  762 

.28  627 

51 

10 

9.71  393 

9.93  230 

9.78  163 

0.21  837 

0.06  770 

0.28  607 

50 

11 

.71  414 

.93  223 

.78  192 

.21  808 

.06  777 

.28  586 

49 

12 

.71  435 

.93  215 

.78  220 

.21  780 

.06  785 

.28  565 

48 

13 

.71  456 

.93  207 

.78  249 

.21  751 

.06  793 

.28  544 

47 

14 

.71  477 

.93  200 

.78  277 

.21  723 

.06  800 

.28  523 

46 

15 

9.71  498 

9.93  192 

^.78  306 

0.21  694 

0.06  808 

0.28  502 

45 

16 

.71  519 

.93  184 

.78  334 

.21  666 

.06  816 

.28  481 

44 

17 

.71  539 

.93  177 

.78  363 

.21  637 

.06  823 

.28  461 

43 

18 

.71  560 

.93  169 

.78  391 

.21  609 

.06  831 

.28  440 

42 

19 

.71  581 

.93  161 

.78  419 

.21  581 

.06  839 

.28  419 

41 

20 

9.71  602 

9.93  154 

.78  448 

0.21  552 

0.06  846 

0.28  398 

40 

21 

.71  622 

.93  146 

.78  476 

.21  524 

.06  854 

.28  378 

39 

22 

.71  643 

.93  138 

.78  505 

.21  495 

.06  862 

.28  357 

38 

23 

.71  664 

.93  131 

.78  533 

.21  467 

.06  869 

.28  336 

37 

24 

.71  685 

.93  123 

.78  562 

.21  438 

.06  877 

.28  315 

36 

25 

9.71  705 

9.93  115 

9.78  590 

0.21  410 

0.06  885 

0.28  295 

35 

26 

.71  726 

.93  108 

.78  618 

.21  382 

.06  892 

.28  274 

34 

27 

.71  747 

.93  100 

.78  647 

.21  353 

.06  900 

.28  253 

33 

28 

.71  767 

.93  092 

.78  675 

.21  325 

.06  908 

.28  233 

32 

29 

.71  788 

.93  084 

.78  704 

.21  296 

.06  916 

.28  212 

31 

30 

9.71  809 

9.93  077 

9.78  732 

0.21  268 

0.06  923 

0.28  191 

30 

31 

.71  829 

.93  069 

.78  760 

.21  240 

.06  931 

.28  171 

29 

32 

.71  850 

.93  061 

.78  789 

.21  211 

.06  939 

.28  150 

28 

33 

.71  870 

.93  053 

.78  817 

.21  183 

.06  947 

.28  130 

27 

34 

.71  89^ 

.93  046 

.78  845 

.21  155 

.06  954 

.28  109 

26 

35 

9.71  911 

9.93  038 

9.78  874 

0.21  126 

0.06  962 

0.28  089 

25 

36 

.71  932 

.93  030 

.78  902 

.21  098 

.06  970 

.28  068 

24 

37 

.71  952 

.93  022 

.78  930 

.21  070 

.06  978 

.28  048 

23 

38 

.71  973 

.93  014 

.78  959 

.21  041 

.06  986 

.28  027 

22 

39 

.71  994 

.93  007 

.78  987 

.21  013 

.06  993 

.28  006 

21 

40 

9.72  014 

9.92  999 

9.79  015 

0.20  985 

0.07  001 

0.27  986 

20 

41 

.72  034 

.92  991 

.79  043 

.20  957 

.07  009 

.27  966 

19 

42 

.72  055 

.92  983 

.79  072 

.20  928 

.07  017 

.27  945 

18 

43 

.72  075 

.92  976 

.79  100 

.20  900 

.07  024 

.27  925 

17 

44 

.72  096 

.92  968 

.79  128 

.20  872 

.07  032 

.27  904 

16 

45 

9.72  116 

9.92  960 

9.79  156 

0.20  844 

0.07  040 

0.27  884 

15 

46 

.72  137 

.92  952 

.79  185 

.20  815 

.07  048 

.27  863 

14 

47 

.72  157 

.92  944 

.79  213 

.20  787 

.07  056 

.27  843 

13 

48 

.72  177 

.92  936 

.79  241 

.20  759 

.07  064 

.27  823 

12 

49 

.72  198 

.92  929 

.79  269 

.20  731 

.07  071 

.27  802 

11 

50 

9.72  218 

9.92  921 

9.79  297 

0.20  703 

0.07  079 

0.27  782 

10 

51 

.72  238 

.92  913 

.79  326 

.20  674 

.07  087 

.27  762 

9 

52 

.72  259 

.92  905 

.79  354 

.20  646 

.07  095 

.27  741 

8 

53 

.72  279 

.92  897 

.79  382 

.20  618 

.07  103 

.27  721 

7 

54 

.72  299 

.92  889 

.79  410 

.20  590 

.07  111 

.27  701 

6 

55 

9.72  320 

9.92  881 

9.79  438 

0.20  562 

0.07  119 

0.27  680 

5 

56 

.72  340 

.92  874 

.79  466 

.20  534 

.07  126 

.27  660 

4 

57 

.72  360 

.92  866 

.79  495 

.20  505 

.07  134 

.27  640 

3 

58 

.72  381 

.92  858 

.79  523 

.20  477 

.07  142 

.27  619 

2 

59 

.72  401 

.92  850 

.79  551 

.20  449 

.07  150 

.27  599 

1 

60 

9.72  421 

9.92  842 

9.79  579 

0.20  421 

0.07  158 

0.27  579 

0 

Cos 

Sin 

Cot 

Tan 

Csc 

Sec 

/ 

121°  (301°) 


(238°)  58' 


228 


Table  4.    Trigonometric  Logarithms 


32°  (212°) 

(327°)  147° 

1 

Sin 

Cos 

Tan 

Cot 

Sec 

Csc 

0 

9.72  421 

9.92  842 

9.79  579 

0.20  421 

0.07  158 

0.27  579 

60 

1 

.72  441 

.92  834 

.79  607 

.20  393 

.07  166 

.27  559 

59 

2 

.72  461 

.92  826 

.79  635 

.20  365 

.07  174 

.27  539 

58 

3 

.72  482 

.92  818 

.79  663 

.20  337 

.07  182 

.27  518 

57 

4 

.72  502 

.92  810 

.79  691 

.20  309 

.07  190 

.27  498 

56 

5 

9.72  522 

9.92  803 

9.79  719 

0.20  281 

0.07  197 

0.27  478 

55 

6 

.72  542 

.92  795 

.79  747 

.20  253 

.07  205 

.27  458 

54 

7 

.72  562 

.92  787 

.79  776 

.20  224 

.07  213 

.27  438 

53 

8 

.72  582 

.92  779 

.79  804 

.20  196 

.07  221 

.27  418 

52 

9 

.72  602 

.92  771 

.79  832 

.20  168 

.07  229 

.27  398 

51 

10 

9.72  622 

9.92  763 

9.79  860 

0.20  140 

0.07  237 

0.27  378 

50 

11 

.72  643 

.92  755 

.79  888 

.20  112 

.07  245 

.27  357 

49 

12 

.72  663 

.92  747 

.79  916 

.20  084 

.07  253 

.27  337 

48 

13 

.72  683 

.92  739 

.79  944 

.20  056 

.07  261 

.27  317 

47 

14 

.72  703 

.92  731 

.79  972 

.20  028 

.07  269 

.27  297 

46 

15 

9.72  723 

9.92  723 

9.80  000 

0.20  000 
.19  9/2 

0.07  277 

0.27  277 

45 

16 

.72  743 

.92  715 

.80  028 

.07  285 

.27  257 

44 

17 

.72  763 

.92  707 

.80  056 

.19  944 

.07  293 

.27  237 

43 

18 

•72  783 

.92  699 

.80  084 

.19  916 

.07  301 

.27  217 

42 

19 

.72  803 

.92  691 

.80  112 

.19  888 

.07  309 

.27  197 

41 

20 

9.72  823 

9.92  683 

9.80  140 

0.19  860 

0.07  317 

0.27  177 

40 

21 

.72  843 

.92  675 

.80  168 

.19  832 

.07  325 

.27  157 

39 

22 

.72  863 

.92  667 

.80  195 

.19  805 

.07  333 

.27  137 

38 

23 

.72  883 

.92  659 

.80  223 

.19  777 

.07  341 

.27  117 

37 

24 

.72  902 

.92  651 

.80  251 

.19  749 

.07  349 

.27  098 

36 

25 

9.72  922 

9.92  643 

9.80  279 

0.19  721 

0.07  357 

0.27  078 

35 

26 

.72  942 

.92  635 

.80  307 

.19  693 

.07  365 

.27  058 

34 

27 

.72  962 

.92  627 

.80  335 

.19  665 

.07  373 

.27  038 

33 

28 

.72  982 

.92  619 

.80  363 

.19  637 

.07  381 

.27  018 

32 

29 

.73  002 

.92  611 

.80  391 

.19  609 

.07  389 

.26  998 

31 

30 

9.73  022 

0.92  603 

9.80  419 

0.19  581 

0.07  397 

0.26  978 

30 

31 

.73  041 

.92  595 

.80  447 

.19  553 

.07  405 

.26  959 

29 

32 

•73  061 

.92  587 

.80  474 

.19  526 

.07  413 

.26  939 

28 

33 

.73  081 

.92  579 

.80  502 

.19  498 

.07  421 

.26  919 

27 

34 

.73  101 

.92  571 

.80  530 

.19  470 

.07  429 

•  .26  899 

26 

35 

9.73  121 

9.92  563 

9.80  558 

0.19  442 

0.07  437 

0.26  879 

25 

36 

.73  140 

.92  555 

.80  586 

.19  414 

.07  445 

.26  860 

24 

37 

.73  160 

.92  546 

.80  614 

.19  386 

.07  454 

.26  840 

23 

38 

.73  180 

.92  538 

.80  642 

.19  358 

.07  462 

.26  820 

22 

39 

.73  200 

.92  530 

.80  669 

.19  331 

.07  470 

.26  800 

21 

40 

9.73  219 

9.92  522 

9.80  697 

0.19  303 

0.07  478 

0.26  781 

20 

41 

.73  239 

.92  514 

.80  725 

.19  275 

.07  486 

.26  761 

19 

42 

.73  259 

.92  506 

.80  753 

.19  247 

.07  494 

.26  741 

18 

43 

.73  278 

.92  498 

.80  781 

.19  219 

.07  502 

.26  722 

17 

44 

.73  298 

.92  490 

.80  808 

.19  192 

.07  510 

.26  702 

16 

45 

9.73  318 

9.92  482 

9.80  836 

0.19  164 

0.07  518 

0.26  682 

15 

46 

.73  337 

.92  473 

-.80  864 

.19  136 

.07  527 

.26  663 

14 

47 

.73  357 

.92  465 

.80  892 

.19  108 

.07  535 

.26  643 

13 

48 

.73  377 

.92  457 

.80  919 

.19  081 

.07  543 

.26  623 

12 

49 

.73  396 

.92  449 

.80  947 

.19  053 

.07  551 

.26  604 

11 

50 

9.73  416 

9.92  441 

9.80  975 

0.19  025 

0.07  559 

0.26  584 

10 

51 

.73  435 

.92  433 

.81  003 

.18  997 

.07  567 

.26  565 

9 

52 

.73  455 

.92  425 

.81  030 

.18  970 

.07  575 

.26  545 

8 

53 

.73  474 

.92  416 

.81  058 

.18  942 

.07  584 

.26  526 

7 

54 

.73  494 

.92  408 

.81  086 

.18  914 

.07  592 

.26  506 

6 

55 

9.73  513 

9.92  400 

9.81  113 

0.18  887 

0.07  600 

0.26  487 

5 

56 

.73  533 

.92  392 

.81  141 

.18  859 

.07  608 

.26  467 

4 

57 

.73  552 

.92  384 

.81  169 

.18  831 

.07  616 

.26  448 

3 

58 

.73  572 

.92  376 

.81  196 

.18  804 

.07  624 

.26  428 

2 

59 

.73  591 

.92  367 

.81  224 

.18  776 

.07  633 

.26  409 

1 

60 

9.73  611 

9.92  359 

9.81  252 

0.18  748 

0.07  641 

0.26  389 

0 

Cos 

Sin 

Cot   ' 

Tan 

Csc 

Sec 

' 

122°  C302°) 


C237°)  57' 


Table  4.    Trigonometric  Logarithms 


229 


33°  (213") 

(326°)  146° 

/ 

Sin 

Cos 

Tan 

Cot 

Sec 

Csc 

0 

9.73  611 

9.92  359 

9.81  252 

0.18  748 

0.07  641 

0.26  389 

60 

1 

.73  630 

.92  351 

.81  279 

.18  721 

.07  649 

.26  370 

59 

2 

.73  650 

.92  343 

.81  307 

.18  693 

.07  657 

.26  350 

58 

3 

.73  669 

.92  335 

.81  335 

.18  665 

.07  665 

.26  331 

57 

4 

.73  689 

.92  326 

.81  362 

.18  638 

.07  674 

.26  311 

56 

5 

9.73  708 

9.92  318 

9.81  390 

0.18  610 

0.07  682 

0.26  292 

55 

6 

.73  727 

.92  310 

.81418 

.18  582 

.07  690 

.26  273 

54 

7 

.73  747 

.92  302 

.81  445 

.18  555 

.07  698 

.26  253 

53 

8 

.73  766 

.92  293 

.81  473 

.18  527 

.07  707 

.26  234 

52 

9 

.73  785 

.92  285 

.81  500 

.18  500 

.07  715 

.26  215 

51 

10 

9.73  805 

9.92  277 

9.81  528 

0.18  472 

0.07  723 

0.26  195' 

50 

11 

.73  824 

.92  269 

.81  556 

.18  444 

.07  731 

.26  176 

49 

12 

.73  843 

.92  260 

.81  583 

.18  417 

.07  740 

.26  157 

48 

13 

.73  863 

.92  252 

.81  611 

.18  389 

.07  748 

.26  137 

47 

14 

.73  882 

.92  244 

.81  638 

.18  362 

.07  756 

.26  118 

46 

15 

9.73  901 

9.92  235 

9.81  666 

0.18  334 

0.07  765 

0.26  099 

45 

16 

.73  921 

.92  227 

.81  693 

.18  307 

.07  773 

.26  079 

44 

17 

.73  940 

.92  219 

.81  721 

.18  279 

.07  781 

.26  060 

43 

18 

.73  959 

.92  211 

.81  748 

.18  252 

.07  789 

.26  041 

42 

19 

.73  978 

.92  202 

.81  776 

.18  224 

.07  798 

.26  022 

41 

20 

9.73  997 

9.92  194 

9.81  803 

0.18  197 

0.07  806 

0.26  003 

40 

21 

.74  017 

.92  186 

.81  831 

.18  169 

.07  814 

.25  983 

39 

22 

.74  036 

.92  177 

.81  858 

.18  142 

.07  823 

.25  964 

38 

23 

.74  055 

.92  169 

.81  886 

.18  114 

.07  831 

.25  945 

37 

24 

.74  074 

.92  161 

.81  913 

.18  087 

.07  839 

.25  926 

36 

25 

9.74  093 

9.92  152 

9.81  941 

0.18  059 

0.07  848 

0.25  907 

35 

26 

.74  113 

.92  144 

.81  968 

.18  032 

.07  856 

.25  887 

34 

27 

.74  132 

.92  136 

.81  996 

.18  004 

.07  864 

.25  868 

33 

28 

.74  151 

.92  127 

.82  023 

.17  977 

.07  873 

.25  849 

32 

29 

.74  170 

.92  119 

.82  051 

.17  949 

.07  881 

.25  830 

31 

30 

9.74  189 

9.92  111 

9.82  078 

0.17  922 

0.07  889 

0.25  811 

30 

31 

.74  208 

.92  102 

.82  106 

.17  894 

.07  898 

.25  792 

29 

32 

.74  227 

.92  094 

.82  133 

.17  867 

.07  906 

.25  773 

28 

33 

.74  246 

.92  086 

.82  161 

.17  839 

.07  914 

.25  754 

27 

34 

.74  265 

.92  077 

.82  188 

.17  812 

.07  923 

.25  735 

26 

35 

9.74  284' 

9.92  069 

9.82  215 

0.17  785 

0.07  931 

0.25  716 

25 

36 

.74  303 

.92  060 

.82  243 

.17  757 

.07  940 

.25  697 

24 

37 

.74  322 

.92  052 

.82  270 

.17  730 

.07  948 

.25  678 

23 

38 

.74  341 

.92  044 

.82  298 

.17  702 

.07  956 

.25  659 

22 

39 

.74  360 

.92  035 

.82  325 

.17  675 

.07  965 

.25  640 

21 

40 

9.74  379 

9.92  027 

9.82  352 

0.17  648 

0.07  973 

0.25  621 

20 

41 

.74  398 

.92  018 

.82  380 

.17  620 

.07  982 

.25  602 

19 

42 

.74  417 

.92  010 

.82  407 

.17  593 

.07  990 

.25  583 

18 

43 

.74  436 

.92  002 

.82  435 

.17  565 

.07  998 

.25  564 

17 

44 

.74  455 

.91  993 

.82  462 

.17  538 

.08  007 

.25  545 

16 

45 

9.74  474 

9.91  985 

9.82  489 

0.17  511 

0.08  015 

0.25  526 

15 

46 

.74  493 

.91  976 

.82  517 

.17  483 

.08  024 

.25  507 

14 

47 

.74  512 

.91  968 

.82  544 

.17  456 

.08  032 

.25  488 

13 

48 

.74  531 

.91  959 

.82  571 

.17  429 

.08  041 

.25  469 

12 

49 

.74  549 

.91  951 

.82  599 

.17  401 

.08  049 

.25  451 

11 

50 

9.74  568 

9.91  942 

9.82  626 

0.17  374 

0.08  058 

0.25  432 

10 

51 

.74  587 

.91  934 

.82  653 

.17  347 

.08  066 

.25  413 

9 

52 

.74  606 

.91  925 

.82  681 

.17  319 

.08  075 

.25  394 

8 

53 

.74  625 

.91  917 

.82  708 

.17  292 

.08  083 

.25  375 

7 

54 

.74  644 

.91  908 

.82  735 

.17  265 

.08  092 

.25  356 

6 

55 

9.74  662 

9.91  900 

9.82  762 

0.17  238 

0.08  100 

0.25  338 

5 

56 

.74  681 

.91  891 

.82  790 

.17  210 

.08  109 

.25  319 

^ 

57 

.74  700 

.91  883 

.82  817 

.17  183 

.08  117 

.25  300 

3 

58 

.74  719 

.91  874 

.82  844 

.17  156 

.08  126 

.25  281 

2 

59 

.74  737 

.91  866 

.82  871 

.17  129 

.08  134 

.25  263 

1 

60 

9.74  756 

9.91  857 

9.82  899 

0.17  101 

0.08  143 

0.25  244 

0 

Cos 

Sin 

Cot 

Tan 

Csc 

Sec 

1 

123°  (303°) 


(236°)  56' 


230 


Table  4.    Trigonometric  Logarithms 


34°  (214°) 

(325°)  145° 

' 

Sin 

Cos 

Tan 

Cot 

Sec 

Csc 

0 

9.74  756 

9.91  857 

9.82  899 

0.17  101 

0.08  143 

0.25  244 

60 

1 

.74  775 

.91  849 

.82  926 

.17  074 

.08  151 

.25  225 

59 

2 

.74  794 

.91  840 

.82  953 

.17  047 

.08  160 

.25  206 

58 

3 

.74  812 

.91  832 

.82  980 

.17  020 

.08  168 

.25  188 

57 

4 

.74  831 

.91  823 

.83  008 

.16  992 

.08  177 

.25  169 

56 

5 

9.74  850 

9.91  815 

9.83  035 

0.16  965 

0.08  185 

0.25  150 

55 

6 

.74  868 

.91  806 

.83  062 

.16  938 

.08  194 

.25  132 

54 

7 

.74  887 

.91  798 

.83  089 

.16  911 

.08  202 

.25  113 

53 

8 

.74  906 

.91  789 

.83  117 

.16  883 

.08  211 

.25  094 

52 

9 

.74  924 

.91  781 

.83  144 

.16  856 

.08  219 

.25  076 

51 

10 

9.74  943 

9.91  772 

9.83  171 

0.16  829 

0.08  228 

0.25  057 

50 

11 

.74  961 

.91  763 

.83  198 

.16  802 

.08  237 

.25  039 

49 

12 

.74  980 

.91  755 

.83  225 

.16  775 

.08  245 

.25  020 

48 

13 

.74  999 

.91  746 

.83  252 

.16  748 

.08  254 

.25  001 

47 

14 

.75  017 

.91  738 

.83  280 

•16  720 

.08  262 

.24  983 

46 

15 

9.75  036 

9.91  729 

9.83  307 

0.16  693 

0.08  271 

0.24  964 

45 

16 

.75  054 

.91  720 

.83  334 

.16  666 

.08  280 

.24  946 

44 

17 

.75  073 

.91  712 

.83  361 

.16  639 

.08  288 

.24  927 

43 

18 

.75  091 

.91  703 

.83  388 

.16  612 

.08  297 

.24  909 

42 

19 

.75  110 

.91  695 

.83  415 

.16  585 

.08  305 

.24  890 

41 

20 

9.75  128 

9.91  686 

9.83  442 

0.16  558 

0.08  314 

0.24  872 

40 

21 

.75  147 

.91  677 

.83  470 

.16  530 

.08  323 

.24  853 

39 

22 

.75  165 

.91  669 

.83  497 

.16  503 

.08  331 

.24  835 

38 

23 

.75  184 

.91  660 

.83  524 

.16  476 

.08  340 

.24  816 

37 

24 

.75  202 

.91  651 

.83  551 

.16  449 

.08  349 

.24  798 

36 

25 

9.75  221 

9.91  643 

9.83  578 

0.16  422 

0.08  357 

0.24  779 

35 

26 

.75  239 

.91  634 

.83  605 

.16  395 

.08  366 

.24  761 

34 

27 

.75  258 

.91  625 

.83  632 

.16  368 

.08  375 

.24  742 

33 

28 

.75  276 

.91  617 

.83  659 

.16  341 

.08  383 

.24  724 

32 

29 

.75  294 

.91  608 

.83  686 

.16  314 

.08  392 

.24  706 

31 

30 

9.75  313 

9.91  599 

9.83  713 

0.16  287 

0.08  401 

0.24  687 

30 

31 

.75  331 

.91  591 

.83  740 

.16  260 

.08  409 

.24  669 

29 

32 

.75  350 

.91  582 

.83  768 

.16  232 

.08  418 

.24  650 

28 

33 

.75  368 

.91  573 

.83  795 

.16  205 

.08  427 

.24  632 

27 

34 

.75  386 

.91  565 

.83  822 

.16  178 

.08  435 

.24  614 

26 

35 

9.75  405 

9.91  556 

9.83  849 

0.16  151 

0.08  444 

0.24  595 

25 

36 

.75  423 

.91  547 

.83  876 

.16  124 

.08  453 

.24  577 

24 

37 

.75  441 

.91  538 

.83  903 

.16  097 

.08  462 

.24  559 

23 

38 

.75  459 

.91  530 

.83  930 

.16  070 

.08  470 

.24  541 

22 

39 

.75  478 

.91  521 

.83  957 

.16  043 

.08  479 

.24  522 

21 

40 

9.75  496 

9.91  512 

9.83  984 

0.16  016 

0.08  488 

0.24  504 

20 

41 

.75  514 

.91  504 

.84  011 

.15  989 

.08  496 

.24  486 

19 

42 

.75  533 

.91  495 

.84  038 

.15  962 

.08  505 

.24  467 

18 

43 

.75  551 

.91  486 

.84  065 

.15  935 

.08  514 

.24  449 

17 

44 

.75  569 

.91  477 

.84  092 

.15  908 

.08  523 

.24  431 

16 

45 

9.75  587 

9.91  469 

9.84  119 

0.15  881 

0.08  531 

0.24  413 

15 

46 

.75  605 

.91  460 

;84  146 

.15  854 

.08  540 

.24  395 

14 

47 

.75  624 

.91  451 

.84  173 

.15  827 

.08  549 

.24  376 

13 

48 

.75  642 

.91  442 

.84  200 

.15  800 

.08  558 

.24  358 

12 

49 

.75  660 

.91  433 

.84  227 

.15  773 

.08  567 

.24  340 

11 

50 

9.75  678 

9.91  425 

9.84  254 

0.15  746 

0.08  575 

0.24  322 

10 

51 

.75  696 

.91  416 

.84  280 

.15  720 

.08  584 

.24  304 

9 

52 

.75  714 

.91  407 

.84  307 

.15  693 

.08  593 

.24  286 

8 

53 

.75  733 

.91  398 

.84  334 

.15  666 

.08  602 

.24  267 

7 

54 

.75  751 

.91  389 

.84  361 

.15  639 

.08  611 

.24  249 

6 

55 

9.75  769 

9.91  381 

9.84  388 

0.15  612 

0.08  619 

0.24  231 

5 

56 

.75  787 

.91  372 

.84  415 

.15  585 

.08  628 

.24  213 

4 

57 

.75  805 

.91  363 

.84  442 

.15  558 

.08  637 

.24  195 

3 

58 

-.75  823 

.91  354 

.84  469 

.15  531 

.08  646 

.24  177 

2 

59 

.75  841 

.91  345 

.84  496 

.15  504 

.08  665 

.24  159 

1 

60 

9.75  859 

9.91  336 

9.84  523 

0.15  477 

0.08  664 

0.24  141 

0 

Cos 

Sin 

Cot 

Tan 

Csc 

Sec 

/ 

124°  (304°) 


(235°)  55= 


Table  4.    Trigonometric  Logarithms 


231 


35°  (215°) 

(324°) 

144° 

' 

Sin 

Cos 

Tan 

Cot 

Sec 

Csc 

0 

9.75  859 

9.91  336 

9.84  523 

0.15  477 

0.08  664 

0.24  141 

60 

1 

.75  877 

.91  328 

.84  550 

.15  450 

.08  672 

.24  123 

59 

2 

.75  895 

.91  319 

.84  576 

.15  424  . 

.08  681 

.24  105 

58 

3 

.75  913 

.91  310 

.84  603 

.15  397 

.08  690 

.24  087 

57 

4 

.75  931 

.91  301 

.84  630 

.15  370 

.08  699 

.24  069 

56 

5 

9.75  949 

9.91  292 

9.84  657 

0.15  343 

0.08  708 

0.24  051 

55 

6 

.75  967 

.91  283 

.84  684 

.15  316 

.08  717 

.24  033 

54 

7 

.75  985 

.91  274 

.84  711 

.15  289 

.08  726 

.24  015 

53 

8 

.76  003 

.91  266 

.84  738 

.15  262 

.08  734 

.23  997 

52 

9 

.76  021 

.91  257 

.84  764 

.15  236 

.08  743 

.23  979 

51 

10 

9.76  039 

9.91  248 

9.84  791 

0.15  209 

0.08  752 

0.23  961 

50 

11 

.76  057 

.91  239 

.84  818 

.15  182 

.08  761 

.23  943 

49 

12 

.76  075 

.91  230 

.84  845 

.15  155 

.08  770 

.23  925 

48 

13 

.76  093 

.91  221 

.84  872 

.15  128 

.08  779 

.23  907 

47 

14 

.76  111 

.91  212 

.84  899 

.15  101 

.08  788 

.23  889 

46 

15 

9.76  129 

9.91  203 

9.84  925 

0.15  075 

0.08  797 

0.23  871 

45 

16 

.76  146 

.91  194 

.84  952 

.15  048 

.08  806 

.23  854 

44 

17 

.76  164 

.91  185 

.84  979 

.15  021 

.08  815 

.23  836 

43 

18 

.76  182 

.91  176 

.85  006 

.14  994 

.08  824 

.23  818 

42 

19 

.76  200 

.91  167 

.85  033 

.14  967 

.08  833 

.23  800 

41 

20 

9.76  218 

9.91  158 

9.85  059 

0.14  941 

0.08  842 

0.23  782 

40 

21 

.76  236 

.91  149 

.85  086 

.14  914 

.08  851 

.23  764 

39 

22 

.76  253 

.91  141 

.85  113 

.14  887 

.08  859 

.23  747 

38 

23 

.76  271 

.91  132 

.85  140 

.14  860 

.08  868 

.23  729 

37 

24 

.76  289 

.91  123 

.85  166 

.14  834 

.08  877 

.23  711 

36 

25 

9.76  307 

9.91  114 

9.85  193 

0.14  807 

0.08  886 

0.23  693 

35 

26 

.76  324 

.91  105 

.85  220 

.14  780 

.08  895 

.23  676 

34 

27 

.76  342 

.91  096 

.85  247 

.14  753 

.08  904 

.23  658 

33 

28 

.76  360 

.91  087 

.85  273 

.14  727 

.08  913 

.23  640 

32 

29 

.76  378 

.91  078 

.85  300 

.14  700 

.08  922 

.23  622 

31 

30 

9.76  395 

9.91  069 

9.85  327 

0.14  673 

0.08  931 

0.23  605 

30 

31 

.76  413 

.91  060 

.85  354 

.14  646 

.08  940 

.23  587 

29 

32 

.76  431 

.91  051 

.85  380 

.14  620 

.08  949 

.23  569 

28 

33 

.76  448 

.91  042 

.85  407 

.14  593 

.08  958 

.23  552 

27 

34 

.76  466 

.91  033 

.85  434 

.14  566 

.08  967 

.23  534 

26 

35 

9.76  484 

9.91  023 

9.85  460 

0.14  540 

0.08  977 

0.23  516 

25 

36 

.76  501 

.91  014 

.85  487 

.14  513 

.08  986 

.23  499 

24 

37 

.76  519 

.91  005 

.85  514 

.14  486 

.08  995 

.23  481 

23 

38 

.76  537 

.90  996 

.85  540 

.14  460 

.09  004 

.23  463 

22 

39 

.76  554 

.90  987 

.85  567 

.14  433 

.09  013 

.23  446 

21 

40 

9.76  572 

9.90  978 

9.85  594 

0.14  406 

0.09  022 

0.23  428 

20 

41 

.76  590 

.90  969 

.85  620 

.14  380 

.09  031 

.23  410 

19 

42 

.76  607 

.90  960 

.85  647 

.14  353 

.09  040 

.23  393 

18 

43 

.76  625 

.90  951 

.85  674 

.14  326 

.09  049 

.23  375 

17 

44 

.76  642 

.90  942 

.85  700 

.14  300 

.09  058 

.23  358 

16 

45 

9.76  660 

9.90  933 

9.85  727 

0.14  273 

0.09  067 

0.23  340 

15 

46 

.76  677 

.90  924 

.85  754 

.14  246 

.09  076 

.23  323 

14 

47 

.76  695 

.90  915 

.85  780 

.14  220 

.09  085 

.23  305 

13 

48 

.76  712 

.90  906 

.85  807 

.14  193 

.09  094 

.23  288 

12 

49 

.76  730 

.90  896 

.85  834 

.14  166 

.09  104 

.23  270 

11 

50 

9.76  747 

9.90  887 

9.85  860 

0.14  140 

0.09  113 

0.23  253 

10 

51 

.76  765 

.90  878 

.85  887 

.14  113 

.09  122 

.23  235 

9 

52 

.76  782 

.90  869 

.85  913 

.14  087 

.09  131 

.23  218 

8 

53 

.76  800 

.90  860 

.85  940 

.14  060 

.09  140 

.23  200 

•  7 

54 

.76  817 

.90  851 

.85  967 

.14  033 

.09  149 

.23  183 

6 

55 

9.76  835 

9.90  842 

9.85  993 

0.14  007 

0.09  158 

0.23  165 

5 

56 

.76  852 

.90  832 

.86  020 

.13  980 

.09  168 

.23  148 

4 

57 

.76  870 

.90  823 

.86  046 

.13  954 

.09  177 

.23  130 

3 

58 

.76  887 

.90  814 

.86  073 

.13  927 

.09  186 

.23  113 

2 

59 

.76  904 

.90  805 

.86  100 

.13  900 

.09  195 

.23  096 

1 

60 

9.76  922 

9.90  796 

9.86  126 

0.13  874 

0.09  204 

0.23  078 

0 

Cos 

Sin 

Cot 

Tan 

Csc 

Sec 

/ 

125°  (305°) 


(234°)  54= 


232 


Table  4.    Trigonometric  Logarithms 


36°  (216°) 

(323°)  143° 

/ 

Sin 

Cos 

Tan 

Cot 

Sec 

Csc 

0 

9.76  922 

9.90  796 

9.86  126 

0.13  874 

0.09  204 

0.23  078 

60 

1 

.76  939 

.90  787 

.86  153 

.13  847 

.09  213 

.23  061 

59 

2 

.76  957 

.90  777 

.86  179 

.13  821 

.09  223 

.23  043 

58 

3 

.76  974 

.90  768 

.86  206 

.13  794 

.09  232 

.23  026 

57 

4 

.76  991 

.90  759 

.86  232 

.13  768 

.09  241 

.23  009 

56 

5 

9.77  009 

9.90  750 

9.86  259 

0.13  741 

0.09  250 

0.22  991 

55 

6 

.77  026 

.90  741 

.86  285 

.13  715 

.09  259 

.22  974 

54 

7 

.77  043 

.90  731 

.86  312 

.13  688 

.09  269 

.22  957 

53 

8 

.77  061 

.90  722 

.86  338 

.13  662 

.09  278 

.22  939 

52 

9 

.77  078 

.90  713 

.86  365 

.13  635 

.09  287 

.22  922 

51 

10 

9.77  095 

9.90  704 

9.86  392 

0.13  608 

0.09  296 

0.22  905 

50 

11 

.77  112 

.90  694 

.86  418 

.13  582 

.09  306 

.22  888 

49 

12 

.77  130 

.90  685 

.86  445 

.13  555 

.09  315 

.22  870 

48 

13 

.77  147 

.90  676 

.86  471 

.13  529 

.09  324 

.22  853 

47 

14 

.77  164 

.90  667 

.86  498 

.13  502 

.09  333 

.22  836 

46 

15 

9.77  181 

9.90  657 

9.86  524 

0.13  476 

0.09  343 

0.22  819 

45 

16 

.77  199 

.90  648 

.86  551 

.13  449 

.09  352 

.22  801 

44 

17 

.77  216 

.90  639 

.86  577 

.13  423 

.09  361 

.22  784 

43 

18 

.77  233 

.90  630 

.86  603 

.13  397 

.09  370 

.22  767 

42 

19 

.77  250 

.90  620 

.86  630 

.13  370 

.09  380 

.22  750 

41 

2a 

9.77  268 

9.90  611 

9.86  656 

0.13  344 

0.09  389 

0.22  732 

40 

21 

.77  285 

.90  602 

.86  683 

.13  317 

.09  398 

.22  715 

39 

22 

.77  302 

.90  592 

.86  709 

.13  291 

.09  408 

.22  698 

38 

23 

.77  319 

.90  583 

.86  736 

.13  264 

.09  417 

.22  681 

37 

24 

.77  336 

.90  574 

.86  762 

.13  238 

.09  426 

.22  664 

36 

25 

9.77  353 

9.90  565 

9.86  789 

0.13  211 

0.09  435 

0.22  647 

35 

26 

.77  370 

.90  555 

.86  815 

.13  185 

.09  445 

.22  630 

34 

27 

.77  387 

.90  546 

.86  842 

.13  158 

.09  454 

.22  613 

33 

28 

.77  405 

.90  537 

.86  868 

.13  132 

.09  463 

.22  595 

32 

29 

.77  422 

.90  527 

.86  894 

•13  106 

.09  473 

.22  578 

31 

30 

9.77  439 

9.90  518 

9.86  921 

0.13  079 

0.09  482 

0.22  561 

30 

31 

.77  456 

.90  509 

.86  947 

.13  053 

.09  491 

.22  544 

29 

32 

.77  473 

.90  499 

.86  974 

.13  026 

.09  501 

.22  527 

28 

33 

.77  490 

.90  490 

.87  000 

.13  000 

.09  510 

.22  510 

27 

34 

.77  507 

.90  480 

.87  027 

.12  973 

.09  520 

.22  493 

26 

35 

9.77  524 

9.90  471 

9.87  053 

0.12  947 

0.09  529 

0.22  476 

25 

36 

.77  541 

.90  462 

.87  079 

.12  921 

.09  538 

.22  459 

24 

37 

.77  558 

.90  452 

.87  106 

.12  894 

.09  548 

.22  442 

23 

38 

.77  575 

.90  443 

.87  132 

.12  868 

.09  557 

.22  425 

22 

39 

.77  592 

.90  434 

.87  158 

.12  842 

.09  566 

.22  408 

21 

40 

9.77  609 

9.90  424 

9.87  185 

0.12  815 

0.09  576 

0.22  391 

20 

41 

.77  626 

.90  415 

.87  211 

.12  789 

.09  585 

.22  374 

19 

42 

.77  643 

.90  405 

.87  238 

.12  762 

.09  595 

.22  357 

18 

43 

.77  660 

.90  396 

.87  264 

.12  736 

.09  604 

.22  340 

17 

44 

.77  677 

.90  386 

.87  290 

.12  710 

.09  614 

.22  323 

16 

45 

9.77  694 

9.90  377 

9.87  317 

0.12  683 

0.09  623 

0.22  306 

15 

46 

.77  711 

.90  368 

.87  343 

.12  657 

.09  632 

.22  289 

14 

47 

.77  728 

.90  358 

.87  369 

.12  631 

.09  642 

.22  272 

13 

48 

.77  744 

.90  349 

.87  396 

.12  604 

.09  651 

.22  256 

12 

49 

.77  761 

.90  339 

.87  422 

.12  578 

.09  661 

.22  239 

11 

50 

9.77  778 

9.90  330 

9.87  448 

0.12  552 

0.09  670 

0.22  222 

10 

51 

.77  795 

.90  320 

.87  475 

.12  525 

.09  680 

.22  205 

9 

52 

.77  812 

.90  311 

.87  501 

.12  499 

.09  689 

.22  188 

8 

53 

.77  829 

.90  301 

.87  527 

.12  473 

.09  699 

.22  171 

7 

54 

.77  846 

.90  292 

.87  554 

.12  446 

.09  708 

.22  154 

6 

55 

9.77  862 

9.90  282 

9.87  580 

0.12  420 

0.09  718 

0.22  138 

5 

56 

.77  879 

.90  273 

.87  606 

.12  394 

.09  727 

.22  121 

4 

57 

.77  896 

.90  263 

.87  633 

.12  367 

.09  737 

.22  104 

3 

58 

.77  913 

.90  254 

.87  659 

.12  341 

.09  746 

.22  087 

2 

59 

.77  930 

.90  244 

.87  685 

.12  315 

.09  756 

.22  070 

1 

60 

9.77  946 

9.90  235 

9.87  711 

0.12  289 

0.09  765 

0.22  054 

0 

Cos 

Sin 

Cot 

Tan 

Csc 

Sec 

/ 

126°  (306°) 


(233°)  53^ 


Table  4.    Trigonometric  Logarithms 


233 


37°  (21 

7°) 

(322°)  142° 

/ 

Sin 

Cos 

Tan 

Cot 

Sec 

Csc 

0 

9.77  946 

9.90  235 

9.87  711 

0.12  289 

0.09  765 

0.22  054 

60 

1 

.77  963 

.90  225 

.87  738 

.12  262 

.09  775 

.22  037 

59 

2 

.77  980 

.90  216 

.87  764 

.12  236 

.09  784 

.22  020 

58 

3 

.77  997 

.90  206 

.87  790 

.12  210 

.09  794 

.22  003 

57 

4 

.78  013 

.90  197 

.87  817 

.12  183 

.09  803 

.21  987 

56 

5 

9.78  030 

9.90  187 

9.87  843 

0.12  157 

0.09  813 

0.21  970 

55 

6 

.78  047 

.90  178 

.87  869 

.12  131 

.09  822 

.21  953 

54 

7 

.78  063 

.90  168 

.87  895 

.12  105 

.09  832 

.21  937 

53 

8 

.78  080 

.90  159 

.87  922 

.12  078 

.09  841 

.21  920 

52 

9 

.78  097 

.90  149 

.87  948 

.12  052 

.09  851 

.21  903 

51 

10 

9.78  113 

9.90  139 

9.87  974 

0.12  026 

0.09  861 

0.21  887 

50 

11 

.78  130 

.90  130 

.88  000 

.12  000 

.09  870 

.21  870 

49 

12 

.78  147 

.90  120 

.88  027 

.11  973 

.09  880 

.21  853 

48 

13 

.78  163 

.90  111 

.88  053 

.11  947 

.09  889 

.21  837 

47 

14 

.78  180 

.90  101 

.88  079 

.11  921 

.09  899 

.21  820 

46 

15 

9.78  197 

9.90  091 

9.88  105 

0.11  895 

0.09  909 

0.21  803 

45 

16 

.78  213 

.90  082 

.88  131 

.11  869 

.09  918 

.21  787 

44 

17 

.78  230 

.90  072 

.88  158 

.11  842 

.09  928 

.21  770 

43 

18 

.78  246 

.90  063 

.88  184 

.11  816 

.09  937 

.21  754 

■42 

19 

.78  263 

.90  053 

.88  210 

.11  790 

.09  947 

.21  737 

41 

20 

9.78  280 

9.90  043 

9.88  236 

0.11  764 

0.09  957 

0.21  720 

40 

21 

.78  296 

.90  034 

.88  262 

.11  738 

.09  966 

.21  704 

39 

22 

.78  313 

.90  024 

.88  289 

.11  711 

.09  976 

.21  687 

38 

23 

.78  329 

.90  014 

.88  315 

.11  685 

.09  986 

.21  671 

37 

24 

.78  346 

.90  005 

.88  341 

.11  659 

.09  995 

.21  654 

36 

25 

9.78  362 

9.89  995 

9.88  367 

0.11  633 

0.10  005 

0.21  638 

35 

26 

.78  379 

.89  985 

.88  393 

.11  607 

.10  015 

.21  621 

34 

27 

.78  395 

.89  976 

.88  420 

.11  580 

.10  024 

.21  605 

33 

28 

.78  412 

.89  966 

.88  446 

.11  554 

.10  034 

.21  588 

32 

29 

.78  428 

.89  956 

.88  472 

.11  528 

.10  044 

.21  572 

31 

30 

9.78  445 

9.89  947 

9.88  498 

0.11  502 

0.10  053 

0.21  555 

30 

31 

.78  461 

.89  937 

.88  524 

.11476 

.10  063 

.21  539 

29 

32 

.78  478 

.89  927 

.88  550 

.11  450 

.10  073 

.21  522 

28 

33 

.78  494 

.89  918 

.88  577 

.11  423 

.10  082 

.21  506 

27 

34 

.78  510 

.89  908 

.88  603 

.11  397 

.10  092 

.21  490 

26 

35 

9.78  527 

9.89  898 

9.88  629 

0.11  371 

0.10  102 

0.21  473 

25 

36 

.78  543 

.89  888 

.88  655 

.11  345 

.10  112 

.21  457 

24 

37 

.78  560 

.89  879 

.88  681 

.11  319 

.10  121 

.21  440 

23 

38 

.78  576 

.89  869 

.88  707 

.11  293 

.10  131 

.21  424 

22 

39 

.78  592 

.89  859 

.88  733 

.11  267 

.10  141 

.21  408 

21 

40 

9.78  609 

9.89  849 

9.88  759 

0.11  241 

0.10  151 

0.21  391 

20 

41 

.78  625 

.89  840 

.88  786 

.11  214 

.10  160 

.21  375 

19 

42 

.78  642 

.89  830 

.88  812 

.11  188 

.10  170 

.21  358 

18 

43 

.78  658 

.89  820 

.88  838 

.11  162 

.10  180 

.21  342 

17 

44 

.78  674 

.89  810 

.88  864 

.11  136 

.10  190 

.21  326 

16 

45 

9.78  691 

9.89  801 

9.88  890 

0.11  110 

0.10  199 

0.21  309 

15 

46 

.78  707 

.89  791 

.88  916 

.11  084 

.10  209 

.21  293 

14 

47 

.78  723 

.89  781 

.88  942 

.11  058 

.10  219 

.21  277 

13 

48 

.78  739 

.89  771 

.88  968 

.11  032 

.10  229 

.21  261 

12 

49 

.78  756 

.89  761 

.88  994 

.11  006 

.10  239 

.21  244 

11 

50 

9.78  772 

9.89  752 

9.89  020 

0.10  980 

0.10  248 

0.21  228 

10 

51 

.78  788 

.89  742 

.89  046 

.10  954 

.10  258 

.21  212 

9 

52 

.78  805 

.89  732 

.89  073 

.10  927 

.10  268 

.21  195 

8 

53 

.78  821 

.89  722 

.89  099 

.10  901 

.10  278 

.21  179 

7 

54 

.78  837 

.89  712 

.89  125 

.10  875 

.10  288 

.21  163 

6 

55 

9.78  853 

9.89  702 

9.89  151 

0.10  849 

0.10  298 

0.21  147 

5 

56 

.78  869 

.89  693 

.89  177 

.10  823 

.10  307 

.21  131 

4 

57 

.78  886 

.89  683 

.89  203 

.10  797 

.10  317 

.21  114 

3 

58 

.78  902 

.89  673 

.89  229 

.10  771 

.10  327 

.21  098 

2 

59 

.78  918 

.89  663 

.89  255 

.10  745 

.10  337 

.21  082 

1 

60 

9.78  934 

9.89  653 

9.89  281 

0.10  719 

0.10  347 

0.21  066 

0 

Cos 

Sin 

Cot 

Tan 

Csc 

Sec 

' 

127°  (307°) 


(232°)  52^ 


234 


Table  4.    Trigonometric  Logarithms 


38°  (218°) 

(321° 

)  141° 

/ 

Sin 

Cos 

Tan 

Cot 

Sec 

Csc 

0 

9.78  934 

9.89  653 

9.89  281 

0.10  719 

0.10  347 

0.21  066 

60 

1 

.78  950 

.89  643 

.89  307 

.10  693 

.10  357 

.21  050 

59 

2 

.78  967 

.89  633 

.89  333 

.10  667 

.10  367 

.21  033 

58 

3 

.78  983 

.89  624 

.89  359 

.10  641 

.10  376 

.21  017 

57 

4 

.78  999 

.89  614 

.89  385 

.10  615 

.10  386 

.21  001 

56 

5 

9.79  015 

9.89  604 

9.89  411 

0.10  589 

0.10  396 

0.20  985 

55 

6 

.79  031 

.89  594 

.89  437 

.10  563 

.10  406 

.20  969 

54 

7 

.79  047 

.89  584 

.89  463 

.10  537 

.10  416 

.20  953 

53 

8 

.79  063 

.89  574 

.89  489 

.10  511 

.10  426 

.20  937 

52 

9 

.79  079 

.89  564 

.89  515 

.10  485 

.10  436 

.20  921 

51 

10 

9.79  095 

9.89  554 

9.89  541 

0.10  459 

0.10  446 

0.20  905 

50 

11 

.79  111 

.89  544 

.89  567 

.10  433 

.10  456 

.20  889 

49 

12 

.79  128 

.89  534 

.89  593 

.10  407 

.10  466 

.20  872 

48 

13 

.79  144 

.89  524 

.89  619 

.10  381 

.10  476 

.20  856 

47 

14 

.79  160 

.89  514 

.89  645 

.10  355 

.10  486 

.20  840 

46 

15 

9.79  176 

9.89  504 

9.89  671 

0.10  329 

0.10  496 

0.20  824 

45 

16 

.79  192 

.89  495 

.89  697 

.10  303 

.10  505 

.20  808 

44 

17 

.79  208 

.89  485 

.89  723 

.10  277 

.10  515 

.20  792 

43 

18 

.79  224 

.89  475 

.89  749 

.10  251 

.10  525 

.20  776 

42 

19 

.79  240 

.89  465 

.89  775 

.10  225 

.10  535 

.20  760 

41 

20 

9.79  256 

9.89  455 

9.89  801 

0.10  199 

0.10  545 

0.20  744 

40 

21 

.79  272 

.89  445 

.89  827 

.10  173 

.10  555 

.20  728 

39 

22 

.79  288 

.89  435 

.89  853 

.10  147 

.10  565 

.20  712 

38 

23 

.79  304 

.89  425 

.89  879 

.10  121 

.10  575 

.20  696 

37 

24 

.79  319 

.89  415 

.89  905 

.10  095 

.10  585 

.20  681 

36 

25 

9.79  335 

9.89  405 

9.89  931 

0.10  069 

0.10  595 

0.20  665 

35 

26 

.79  351 

.89  395 

.89  957 

.10  043 

.10  605 

.20  649 

34 

27 

.79  367 

.89  385 

.89  983 

.10  017 

.10  615 

.20  633 

33 

28 

.79  383 

.89  375 

.90  009 

.09  991 

.10  625 

.20  617 

32 

29 

.79  399 

.89  364 

.90  035 

.09  965 

.10  636 

.20  601 

31 

30 

9.79  415 

9.89  354 

9.90  061 

0.09  939 

0.10  646 

0.20  585 

30 

31 

.79  431 

.89  344 

.90  086 

.09  914 

.10  656 

.20  569 

29 

32 

.79  447 

.89  334 

.90  112 

.09  888 

.10  666 

.20  553 

28 

33 

.79  463 

.89  324 

.90  138 

.09  862 

.10  676 

.20  537 

27 

34 

.79  478 

.89  314 

.90  164 

.09  836 

.10  686 

.20  522 

26 

35 

9.79  494 

9.89  304 

9.90  190 

0.09  810 

0.10  696 

0.20  506 

25 

36 

.79  510 

.89  294 

.90  216 

.09  784 

.10  706 

.20  490 

24 

37 

.79  526 

.89  284 

.90  242 

.09  758 

.10  716 

.20  474 

23 

38 

.79  542 

.89  274 

.90  268 

.09  732 

.10  726 

.20  458 

22 

39 

.79  558 

.89  264 

.90  294 

.09  706 

.10  736 

.20  442 

21 

40 

9.79  573 

9.89  254 

9.90  320 

0.09  680 

0-10  746 

0.20  427 

20 

41 

.79  589 

.89  244 

.90  346 

.09  654 

.10  756 

.20  411 

19 

42 

.79  605 

.89  233 

.90  371 

.09  629 

.10  767 

.20  395 

18 

43 

.79  621 

.89  223 

.90  397 

.09  603 

.10  777 

.20  379 

17 

44 

.79  636 

.89  213 

.90  423 

.09  577 

.10  787 

.20  364 

16 

45 

9.79  652 

9.89  203 

9.90  449 

0.09  551 

0.10  797 

0.20  348 

15 

46 

.79  668 

.89  193 

.90  475 

.09  525 

.10  807 

.20  332 

14 

47 

.79  684 

.89  183 

.90  501 

.09  499 

.10  817 

.20  316 

13 

48 

.79  699 

.89  173 

.90  527 

.09  473 

.10  827 

.20  301 

12 

49 

.79  715 

.89  162 

.90  553 

.09  447 

.10  838 

.20  285 

11 

50 

9.79  731 

9.89  152 

9.90  578 

0.09  422 

0.10  848 

0.20  269 

10 

51 

.79  746 

.89  142 

.90  604 

.09  396 

.10  858 

.20  254 

9 

52 

.79  762 

.89  132 

.90  630 

.09  370 

.10  868 

.20  238 

8 

53 

.79  778 

.89  122 

.90  656 

.09  344 

.10  878 

.20  222 

7 

54 

.79  793 

.89  112 

.90  682 

.09  318 

.10  888 

.20  207 

6 

55 

9.79  809 

9.89  101 

9.90  708 

0.09  292 

0.10  899 

0.20  191 

5 

56 

.79  825 

.89  091 

.90  734 

.09  266 

.10  909 

.20  175 

4 

57 

.79  840 

.89  081 

.90  759 

.09  241 

.10  919 

.20  160 

3 

58 

.79  856 

.89  071 

.90  785 

.09  215 

.10  929 

.20  144 

2 

59 

.79  872 

.89  060 

.90  811 

.09  189 

.10  940 

.20  128 

1 

60 

9.79  887 

9.89  050 

9.90  837 

0.09  163 

0.10  950 

0.20  113 

0 

Cos 

Sin 

Cot 

Tan 

Csc 

Sec 

/ 

128°  (308°) 


(231°)  51= 


Table  4.    Trigonometric  LogJirithms 


235 


39°  (219°) 

(320°)  140° 

' 

Sin 

Cos 

Tan 

Cot 

Sec 

Csc 

0 

9.79  887 

9.89  050 

9.90  837 

0.09  163   0 

10  950 

0.20  113 

60 

1 

.79  903 

.89  040 

.90  863 

.09  137 

10  960 

.20  097 

59 

2 

.79  918 

.89  030 

.90  889 

.09  111 

10  970 

.20  082 

58 

3 

.79  934 

.89  020 

.90  914 

.09  086 

10  980 

.20  066 

57 

4 

.79  950 

.89  009 

.90  940 

.09  060 

10  991 

.20  050 

56 

5 

9.79  965 

9.88  999 

9.90  966 

0.09  034   0 

11  001 

0.20  035 

55 

6 

.79  981 

.88  989 

.90  992 

.09  008 

11  Oil 

.20  019 

54 

7 

.79  996 

.88  978 

.91  018 

.08  982 

11  022 

.20  004 

53 

8 

.80  012 

.88  968 

.91  043 

.08  957 

11  032 

.19  988 

52 

9 

.80  027 

.88  958 

.91  069 

.08  931 

11  042 

.19  973 

51 

10 

9.80  043 

9.88  948 

9.91  095 

0.08  905   0 

11  052 

0.19  957 

50 

11 

.80  058 

.88  937 

.91  121 

.08  879 

1 1  063 

.19  942 

49 

12 

.80  074 

.88  927 

.91  147 

.08  853 

11  073 

.19  926 

48 

13 

.80  089 

.88  917 

.91  172 

.08  828 

11  083 

.19  911 

47 

14 

.80  105 

.88  906 

.91  198 

.08  802 

11  094 

.19  895 

46 

15 

9.80  120 

9.88  896 

9.91  224 

0.08  776   0 

11  104 

0.19  880 

45 

16 

.80  130 

.88  886 

.91  250 

.08  750 

11  114 

.19  864 

44 

17 

.80  151 

.88  875 

.91  276 

.08  724 

11  125 

.19  849 

43 

18 

.80  166 

.88  865 

.91  301 

.08  699 

11  135 

.19  834 

42 

19 

.80  182 

.88  855 

.91  327 

.08  673 

11  145 

.19  818 

41 

20 

9.80  197 

9.88  844 

9.91  353 

0.08  647   0 

11  156 

0.19  803 

40 

21 

.80  213 

.88  834 

.91  379 

.08  621 

11  166 

.19  787 

39 

22 

.80  228 

.88  824 

.91  404 

.08  596 

11  176 

.19  772 

38 

23 

.80  244 

.88  813 

.91  430 

.08  570 

11  187 

.19  756 

37 

24 

.80  259 

.88  803 

.91  456 

.08  544 

11  197 

.19  741 

36 

25 

9.80  274 

9.88  793 

9.91  482 

0.08  518   0 

11  207 

0.19  726 

35 

26 

.80  290 

.88  782 

.91  507 

.08  493 

11  218 

.19  710 

34 

27 

.80  305 

.88  772 

.91  533 

.08  467 

11  228 

.19  695 

33 

28 

.80  320 

.88  761 

.91  559 

.08  441 

11  239 

.19  680 

32 

29 

.80  336 

.88  751 

.91  585 

.08  415 

11  249 

.19  664 

31 

30 

9.80  351 

9.88  741 

9.91  610 

0.08  390   0 

11  259 

0.19  649 

30 

31 

.80  366 

.88  730 

.91  636 

.08  364 

11  270 

.19  634 

29 

32 

.80  382 

.88  720 

.91  662 

.08  338 

11  280 

.19  618 

28 

33 

.80  397 

.88  709 

.91  688 

.08  312 

11  291 

.19  603 

27 

34 

.80  412 

.88  699 

.91  713 

.08  287 

11  301 

.19  588 

26 

35 

9.80  428 

9.88  688 

9.91  739 

0.08  261   0 

11  312 

0.19  572 

25 

36 

.80  443 

.88  678 

.91  765 

.08  235 

11  322 

.19  557 

24 

37 

.80  458 

.88  668 

.91  791 

.08  209 

11  332 

.19  542 

23 

38 

.80  473 

.88  657 

.91  816 

.08  184 

11  343 

.19  527 

22 

39 

.80  489 

.88  647 

.91  842 

.08  158 

11  353 

.19  511 

21 

40 

9.80  504 

9.88  636 

9.91  868 

0.08  132   0 

11  364 

0.19  496 

20 

41 

.80  519 

.88  626 

.91  893 

.08  107 

11  374 

.19  481 

19 

42 

.80  534 

.88  615 

.91  919 

.08  081 

11  385 

.19  466 

18 

43 

.80  550 

.88  605 

.91  945 

.08  055 

11  395 

.19  450 

17 

44 

.80  565 

.88  594 

.91  971 

.08  029 

11406 

.19  435 

16 

45 

9.80  580 

9.88  584 

9.91  996 

0.08  004   0 

11  416 

0.19  420 

15 

46 

.80  595 

.88  573 

.92  022 

.07  978 

11427 

.19  405 

14 

47 

.80  610 

.88  563 

.92  048 

.07  952 

11  437 

.19  390 

13 

48 

.80  625 

.88  552 

.92  073 

.07  927 

11  448 

.19  375 

12 

49 

.80  641 

.88  542 

.92  099 

.07  901 

11458 

.19  359 

11 

50 

9.80  656 

9.88  531 

9.92  125 

0.07  875   0 

11  469 

0.19  344 

10 

51 

.80  671 

.88  521 

.92  150 

.07  850 

11  479 

.19  329 

9 

52 

.80  686 

.88  510 

.92  176 

.07  824 

11  490 

.19  314 

8 

53 

.80  701 

.88  499 

.92  202 

.07  798 

11  501 

.19  299 

7 

54 

.80  716 

.88  489 

.92  227 

.07  773 

11511 

.19  284 

6 

55 

9.80  731 

9.88  478 

9.92  253 

0.07  747   0 

11522 

0.19  269 

5 

56 

.80  746 

.88  468 

.92  279 

.07  721 

11  532 

.19  254 

4 

57 

.80  762 

.88  457 

.92  304 

.07  696 

11543 

.19  238 

3 

58 

.80  777 

.88  447 

.92  330 

.07  670 

11553 

.19  223 

2 

59 

.80  792 

.88  436 

.92  356 

.07  644 

11  564 

.19  208 

1 

60 

9.80  807 

9.88  425 

9.92  381 

0.07  619   0 

11  575 
Csc 

0.19  193 

0 

Cos 

Sin 

Cot 

Tan 

Sec 

' 

129°  (309°) 


(230°)  50^ 


236 


Table  4.    Trigonometric  Logarithms 


40°  (220°) 

(319° 

)  139° 

/ 

Sin 

Cos 

Tan 

Cot 

Sec 

Csc 

0    9 

80  807 

9.88  425 

9.92  381 

0.07  619   0 

11  575 

0.19  193 

60 

1 

80  822 

.88  415 

.92  407 

.07  593 

11585 

.19  178 

59 

2 

80  837 

.88  404 

.92  433 

.07  567 

11  596 

.19  163 

58 

3 

80  852 

.88  394 

.92  458 

.07  542 

11606 

.19  148 

57 

4 

80  867 

.88  383 

.92  484 

.07  516 

11  617 

.19  133 

56 

5    9 

80  882 

9.88  372 

9.92  510 

0.07  490   0 

11  628 

0.19  118 

55 

6 

80  897 

.88  362 

.92  535 

.07  465 

11  638 

.19  103 

54 

7 

80  912 

.88  351 

.92  561 

.07  439 

11  649 

.19  088 

53 

8 

80  927 

.88  340 

.92  587 

.07  413 

11  660 

.19  073 

52 

9 

80  942 

.88  330 

.92  612 

.07  388 

11  670 

.19  058 

51 

10    9 

80  957 

9.88  319 

9.92  638 

0.07  362   0 

11681 

0.19  043 

50 

11 

80  972 

.88  308 

.92  663 

.07  337 

11  692 

.19  028 

49 

12 

80  987 

.88  298 

.92  689 

.07  311 

11  702 

.19  013 

48 

13 

81  002 

.88  287 

.92  715 

.07  285 

11  713 

.18  998 

47 

14 

81  017 

.88  276 

.92  740 

•07  260 

11  724 

.18  983 

46 

15    9 

81  032 

9.88  266 

9.92  766 

0.07  234   0 

11  734 

0.18  968 

45 

16 

81  047 

.88  255 

.92  792 

.07  208 

11  745 

.18  953 

44 

17 

81  061 

.88  244 

.92  817 

.07  183 

11  756 

.18  939 

43 

18 

81  076 

.88  234 

.92  843 

.07  157 

11  766 

.18  924 

42 

19 

81  091 

.88  223 

.92  868 

•07  132 

11  777 

.18  909 

41 

20    9 

81  106 

9.88  212 

9.92  894 

0.07  106   0 

11  788 

0.18  894 

40 

21 

81  121 

.88  201 

.92  920 

.07  080 

11  799 

.18  879 

39 

22 

81  136 

.88  191 

.92  945 

.07  055 

11  809 

.18  864 

38 

23 

81  151 

.88  180 

.92  971 

.07  029 

11  820 

.18  849 

37 

24 

81  166 

.88  169 

.92  996 

.07  004 

11  831 

.18  834 

36 

25    9 

81  180 

9.88  158 

9.93  022 

0.06  978  0 

11  842 

0.18  820 

35 

26 

81  195 

.88  148 

.93  048 

.06  952 

11  852 

.18  805 

34 

27 

81  210 

.88  137 

.93  073 

.06  927 

11  863 

.18  790 

33 

28 

81  225 

.88  126 

.93  099 

.06  901 

11  874 

.18  775 

32 

29 

81  240 

.88  115 

.93  124 

.06  876 

11  885 

.18  760 

31 

30    9 

81  254 

9.88  105 

9.93  150 

0.06  850  0 

11  895 

0.18  746 

30 

31 

81  269 

.88  094 

.93  175 

.06  825 

11  906 

.18  731 

29 

32 

81  284 

.88  083 

.93  201 

.06  799 

11  917 

.18  716 

28 

33 

81  299 

.88  072 

.93  227 

.06  773 

11  928 

.18  701 

27 

34 

81  314 

.88  061 

.93  252 

.06  748 

11  939 

.18  686 

26 

35    9 

81  328 

9.88  051 

9.93  278 

0.06  722   0 

11  949 

0.18  672 

25 

36 

81  343 

.88  040 

.93  303 

.06  697 

11  960 

.18  657 

24 

37 

81  358 

.88  029 

.93  329 

.06  671 

11  971 

.18  642 

23 

38 

81  372 

.88  018 

.93  354 

.06  646 

11  982 

.18  628 

22 

39 

81387 

.88  007 

.93  380 

.06  620 

11  993 

.18  613 

21 

40    9 

81  402 

9.87  996 

9.93  406 

0.06  594   0 

12  004 

0.18  598 

20 

41 

81  417 

.87  985 

.93  431 

.06  569 

12  015 

.18  583 

19 

42 

81  431 

.87  975 

.93  457 

.06  543 

12  025 

.18  569 

18 

43 

81  446 

.87  964 

.92  482 

.06  518 

12  036 

.18  554 

17 

44 

81  461 

.87  953 

.93  508 

.06  492 

12  047 

.18  539 

16 

45    9 

81475 

9.87  942 

9.03  533 

0.06  467   0 

12  058 

0.18  525 

15 

46 

81  490 

.87  931 

.93  559 

.06  441 

12  069 

.18  510 

14 

47 

81  505 

.87  920 

.93  584 

.06  416 

12  080 

.18  495 

13 

48 

81  519 

.87  909 

.93  610 

.06  390 

12  091 

.18  481 

12 

49 

81534 

.87  898 

.93  636 

.06  364 

12  102 

.18  466 

11 

50    9 

81  549 

9.87  887 

9.93  661 

0.06  339   0 

12  113 

0.18  451 

10 

51 

81  563 

.87  877 

.93  687 

.06  313 

12  123 

.18  437 

9 

52 

81  578 

.87  866 

.93  712 

.06  288 

12  134 

.18  422 

8 

53 

81  592 

.87  855 

.93  738 

.06  262 

12  145 

.18  408 

7 

54 

81  607 

.87  844 

.93  763 

.06  237 

12  156 

.18  393 

6 

55    9 

81  622 

9.87  833 

9.93  789 

0.06  211   0 

12  167 

0.18  378 

5 

56 

81  636 

.87  822 

.93  814 

.06  186 

12  178 

.18  364 

4 

57 

81  651 

.87  811 

.93  840 

.06  160 

12  189 

.18  349 

3 

58 

81  665 

.87  800 

.93  865 

.06  135 

12  200 

.18  335 

2 

59 

81  680 

.87  789 

.93  891 

.06  109 

12  211 

.18  320 

1 

60    9 

81  694 

9.87  778 

9.93  916 

0.06  084   0 

12  222 
Csc 

.18  306 

0 

Cos 

Sin 

Cot 

Tan 

Sec 

/ 

130°  (310°) 


(229°)  49' 


Table  4.    Trigonometric  Logarithms 


237 


41°  (221°) 

(318° 

)  138° 

/ 

Sin 

Cos 

Tan 

Cot 

Sec 

Csc 

0 

9.81  694 

9.87  778 

9.93  916 

0.06  084 

0.12  222 

0.18  306 

60 

1 

.81  709 

.87  767 

.93  942 

.06  058 

.12  233 

.18  291 

59 

2 

.81  723 

.87  756 

.93  967 

.06  033 

.12  244 

.18  277 

58 

3 

.81  738 

.87  745 

.93  993 

.06  007 

.12  255 

.18  262 

57 

4 

.81  752 

.87  734 

.94  018 

.05  982 

.12  266 

.18  248 

56 

5 

9.81  767 

9.87  723 

9.94  044 

0.05  956 

0.12  277 

0.18  233 

55 

6 

.81  781 

.87  712 

.94  069 

.05  931 

.12  288 

.18  219 

54 

7 

.81  796 

.87  701 

.94  095 

.05  905 

.12  299 

.18  204 

53 

8 

.81  810 

.87  690 

.94  120 

.05  880 

.12  310 

.18  190 

52 

9 

.81  825 

.87  679 

.94  146 

.05  854 

.12  321 

.18  175 

51 

10 

9.81  839 

9.87  668 

9.94  171 

0.05  829 

0.12  332 

0.18  161 

50 

11 

.81  854 

.87  657 

.94  197 

.05  803 

.12  343 

.18  146 

49 

12 

.81  868 

.87  646 

.94  222 

.05  778 

.12  354 

.18  132 

48 

13 

.81  882 

.87  635 

.94  248 

.05  752 

.12  365 

.18  118 

47 

14 

.81  897 

.87  624 

.94  273 

.05  727 

.12  376 

.18  103 

46 

15 

9.81  911 

9.87  613 

9.94  299 

0.05  701 

0.12  387 

0.18  089 

45 

16 

.81  926 

.87  601 

.94  324 

.05  676 

.12  399 

.18  074 

44 

17 

.81  940 

.87  590 

.94  350 

.05  650 

.12  410 

.18  060 

43 

18 

.81  955 

.87  579 

.94  375 

.05  625 

.12  421 

.18  045 

42 

19 

.81  969 

.87  568 

.94  401 

.05  599 

.12  432 

.18  031 

41 

20 

9.81  983 

9.87  557 

9.94  426 

0.05  574 

0.12  443 

0.18  017 

40 

21 

.81  998 

.87  546 

.94  452 

.05  548 

.12  454 

.18  002 

39 

22 

.82  012 

.87  535 

.94  477 

.05  523 

.12  465 

.17  988 

38 

23 

.82  026 

.87  524 

.94  503 

.05  497 

.12  476 

.17  974 

37 

24 

.82  041 

.87  513 

.94  528 

.05  472 

.12  487 

.17  959 

36 

25 

9.82  055 

9.87  501 

9.94  554 

0.05  446 

0.12  499 

0.17  945 

35 

26 

.82  069 

.87  490 

.94  579 

.05  421 

.12  510 

.17  931 

34 

27 

.82  084 

.87  479 

.94  604 

.05  396 

.12  521 

.17  916 

33 

28 

.82  098 

.87  468 

.94  630 

.05  370 

.12  532 

.17  902 

32 

29 

.82  112 

.87  457 

.94  655 

.05  345 

.12  543 

.17  888 

31 

30 

9.82  126 

9.87  446 

9.94  681 

0.05  319 

0.12  554 

0.17  874 

30 

31 

.82  141 

.87  434 

.94  706 

.05  294 

.12  566 

.17  859 

29 

32 

.82  155 

.87  423 

.94  732 

.05  268 

.12  577 

.17  845 

28 

33 

.82  169 

.87  412 

.94  757 

.05  243 

.12  588 

.17  831 

27 

34 

.82  184 

.87  401 

.94  783 

•05  217 

.12  599 

.17  816 

26 

35 

9.82  198 

9.87  390 

9.94  808 

0.05  192 

0.12  610 

0.17  802 

25 

36 

.82  212 

.87  378 

.94  834 

.05  166 

.12  622 

.17  788 

24 

37 

.82  226 

.87  367 

.94  859 

.05  141 

.12  633 

.17  774 

23 

38 

.82  240 

.87  356 

.94  884 

.05  116 

.12  644 

.17  760 

22 

39 

.82  255 

.87  345 

.94  910 

.05  090 

.12  655 

.17  745 

21 

40 

9.82  269 

9.87  334 

9.94  935 

0.05  065 

0.12  666 

0.17  731 

20 

41 

.82  283 

.87  322 

.94  961 

.05  039 

.12  678 

.17  717 

19 

42 

.82  297 

.87  311 

.94  986 

.05  014 

.12  689 

.17  703 

18 

43 

.82  311 

.87  300 

.95  012 

.04  988 

.12  700 

.17  689 

17 

44 

.82  326 

.87  288 

.95  037 

.04  963 

.12  712 

.17  674 

16 

45 

9.82  340 

9.87  277 

9.95  062 

0.04  938 

0.12  723 

0.17  660 

15 

46 

.82  354 

.87  266 

.95  088 

.04  912 

.12  734 

.17  646 

14 

47 

.82  368 

.87  255 

.95  113 

.04  887 

.12  745 

.17  632 

13 

48 

•82  382 

.87  243 

.95  139 

.04  861 

.12  757 

.17  618 

12 

49 

.82  396 

.87  232 

.95  164 

.04  836 

.12  768 

.17  604 

11 

50 

9.82  410 

9.87  221 

9.95  190 

0.04  810 

0.12  779 

0.17  590 

10 

51 

.82  424 

.87  209 

.95  215 

.04  785 

.12  791 

.17  576 

9 

52 

.82  439 

.87  198 

.95  240 

.04  760 

.12  802 

.17  561 

8 

53 

.82  453 

.87  187 

.95  266 

.04  734 

.12  813 

.17  547 

7 

54 

.82  467 

.87  175 

.95  291 

.04  709 

.12  825 

.17  533 

6 

55 

9.82  481 

9.87  164 

9.95  317 

0.04  683 

0.12  836 

0.17  519 

5 

56 

.82  495 

.87  153 

.95  342 

.04  658 

.12  847 

.17  505 

4 

57 

.82  509 

.87  141 

.95  368 

.04  632 

.12  859 

.17  491 

3 

58 

.82  523 

.87  130 

.95  393 

.04  607 

.12  870 

.17  477 

2 

59 

.82  537 

.87  119 

.95  418 

.04  582 

.12  881 

.17  463 

1 

60 

9.82  551 

9.87  107 

9.95  444 

0.04  556 

0.12  893 

0.17  449 

0 

Cos 

Sin 

Cot 

Tan 

Csc 

Sec 

' 

131°  (31 

1°) 

(228 

')  48° 

238 


Table  4.    Trigonometric  Logarithms 


42''  (222°) 

(317°)  137° 

/ 

Sin 

Cos 

Tan 

Cot 

Sec 

Csc 

0 

9.82  551 

9.87  107 

9.95  444 

0.04  556 

0.12  893 

0.17  449 

60 

1 

.82  565 

.87  096 

.95  469 

.04  531 

.12  904 

.17  435 

59 

2 

.82  579 

.87  085 

.95  495 

.04  505 

.12  915 

.17  421 

58 

3 

.82  593 

.87  073 

.95  520 

.04  480 

.12  927 

.17  407 

57 

4 

.82  607 

.87  062 

.95  545 

.04  455 

.12  938 

.17  393 

56 

5 

9.82  621 

9.87  050 

9.95  571 

0.04  429 

0.12  950 

0.17  379 

55 

6 

.82  635 

.87  039 

.95  596 

.04  404 

.12  961 

.17  365 

54 

7 

.82  649 

.87  028 

.95  622 

.04  378 

.12  972 

.17  351 

53 

8 

.82  663 

.87  016 

.95  647 

.04  353 

.12  984 

.17  337 

52 

9 

.82  677 

.87  005 

.95  672 

.04  328 

.12  995 

.17  323 

51 

10 

9.82  691 

9.86  993 

9.95  698 

0.04  302 

0.13  007 

'  0.17  309 

50 

11 

.82  705 

.86  982 

.95  723 

.04  277 

.13  018 

.17  295 

49 

12 

.82  719 

.86  970 

.95  748 

.04  252 

.13  030 

.17  281 

48 

13 

.82  733 

.86  959 

.95  774 

.04  226 

.13  041 

.17  267 

47 

14 

.82  747 

.86  947 

.95  799 

.04  201 

.13  053 

.17  253 

46 

15 

9.82  761 

9.86  936 

9.95  825 

0.04  175 

0.13  064 

0.17  239 

45 

16 

.82  775 

.86  924 

.95  850 

.04  150 

.13  076 

.17  225 

44 

17 

.82  788 

.86  913 

.95  875 

.04  125 

.13  087 

.17  212 

43 

18 

.82  802 

.86  902 

.95  901 

.04  099 

.13  098 

.17  198 

42 

19 

.82  816 

.86  890 

.95  926 

.04  074 

.13  110 

.17  184 

41 

20 

9.82  830 

9.86  879 

9.95  952 

0.04  048 

0.13  121 

0.17  170 

40 

21 

.82  844 

.86  867 

.95  977 

.04  023 

.13  133 

.17  156 

39 

22 

.82  858 

.86  855 

.96  002 

.03  998 

.13  145 

.17  142 

38 

23 

.82  872 

.86  844 

.96  028 

.03  972 

.13  156 

.17  128 

37 

24 

.82  885 

.86  832 

.96  053 

.03  947 

.13  168 

.17  115 

36 

25 

9.82  899 

9.86  821 

9.96  078 

0.03  922 

0.13  179 

0.17  101 

35 

26 

.82  913 

.86  809 

.96  104 

.03  896 

.13  191 

.17  087 

34 

27 

.82  927 

.86  798 

.96  129 

.03  871 

.13  202 

.17  073 

33 

28 

.82  941 

.86  786 

.96  155 

.03  845 

.13  214 

.17  059 

32 

29 

.82  955 

.86  775 

.96  180 

.03  820 

.13  225 

.17  045 

31 

30 

9.82  968 

9.86  763 

9.96  205 

0.03  795 

0.13  237 

0.17  032 

30 

31 

.82  982 

.86  752 

.96  231 

.03  769 

.13  248 

.17  018 

29 

32 

.82  996 

.86  740 

.96  256 

.03  744 

.13  260 

.17  004 

28 

33 

.83  010 

.86  728 

.96  281 

.03  719 

.13  272 

.16  990 

27 

34 

.83  023 

.86  717 

.96  307 

.03  693 

.13  283 

.16  977 

26 

35 

9.83  037 

9.86  705 

9.96  332 

0.03  668 

0.13  295 

0.16  963 

25 

36 

.83  051 

.86  694 

.96  357 

.03  643 

.13  306 

.16  949 

24 

37 

.83  065 

.86  682 

.96  383 

.03  617 

.13  318 

.16  935 

23 

38 

.83  078 

.86  670 

.96  408 

.03  592 

.13  330 

.16  922 

22 

39 

.83  092 

.86  659 

.96  433 

.03  567 

.13  341 

.16  908 

21 

40 

9.83  106 

9.86  647 

9.96  459 

0.03  541 

0.13  353 

0.16  894 

20 

41 

.83  120 

.86  635 

.96  484 

.03  516 

.13  365 

.16  880 

19 

42 

.83  133 

.86  624 

.96  510 

.03  490 

.13  376 

.16  867 

18 

43 

.83  147 

.86  612 

.96  535 

.03  465 

.13  388 

.16  853 

17 

44 

.83  161 

.86  600 

.96  560 

.03  440 

.13  400 

.16  839 

16 

45 

9.83  174 

9.86  589 

9,96  586 

0.03  414 

0.13  411 

0.16  826 

15 

46 

.83  188 

.86  577 

.96  611 

.03  389 

.13  423 

.16  812 

14 

47 

.83  202 

.86  565 

.96  636 

.03  364 

.13  435 

.16  798 

13 

48 

.83  215 

.86  554 

.96  662 

.03  338 

.13  446 

.16  785 

12 

49 

.83  229 

.86  542 

.96  687 

.03  313 

.13  458 

.16  771 

11 

50 

9.83  242 

9.86  530 

9.96  712 

0.03  288 

0.13  470 

0.16  758 

10 

51 

.83  256 

.86  518 

.96  738 

.03  262 

.13  482 

.16  744 

9 

52 

.83  270 

.86  507 

.96  763 

.03  237 

.13  493 

.16  730 

8 

53 

.83  283 

.86  495 

.96  788 

.03  212 

.13  505 

.16  717 

7 

54 

.83  297 

.86  483 

.96  814 

.03  186 

.13  517 

.16  703 

6 

55 

9.83  310 

9.86  472 

9.96  839 

0.03  161 

0.13  528 

0.16  690 

5 

56 

.83  324 

.86  460 

.96  864 

.03  136 

.13  540 

.16  676 

4 

57 

.83  338 

.86  448 

.96  890 

.03  110 

.13  552 

.16  662 

3 

58 

.83  351 

.86  436 

.96  915 

.03  085 

.13  564 

.16  649 

2 

59 

.83  365 

.86  425 

.96  940 

.03  060 

.13  575 

.16  635 

1 

60 

9.83  378 

9.86  413 

9.96  966 

0.03  034 

0.13  587 

0.16  622 

0 

Cos 

Sin 

Cot 

Tan 

Csc 

Sec 

1 

132°  (312°) 


(227°)  47= 


Table  4.    Trigonometric  Logarithms 


239 


43°  (223°) 

(316°)  136° 

' 

Sin 

Cos 

Tan 

Cot 

Sec 

Csc 

0 

9.83  378 

9.86  413 

9.96  966 

0.03  034 

0.13  587 

0.16  622 

60 

1 

.83  392 

.86  401 

.96  991 

.03  009 

.13  599 

.16  608 

59 

2 

.83  405 

.86  389 

.97  016 

.02  984 

.13  611 

.16  595 

58 

3 

.83  419 

.86  377 

.97  042 

.02  958 

.13  623 

.16  581 

57 

4 

.83  432 

.86  366  • 

.97  067 

.02  933 

.13  634 

.16  568 

56 

5 

9.83  446 

9.86  354 

9.97  092 

0.02  908 

0.13  646 

0.16  554 

55 

6 

.83  459 

.86  342 

.97  118 

.02  882 

.13  658 

.16  541 

54 

7 

.83  473 

.86  330 

.97  143 

.02  857 

.13  670 

.16  527 

53 

8 

.83  486 

.86  318 

.97  168 

.02  832 

.13  682 

.16  514 

52 

9 

.83  500 

.86  306 

.97  193 

.02  807 

.13  694 

.16  500 

51 

10 

9.83  513 

9.86  295 

9.97  219 

0.02  781 

0.13  705 

0.16  487 

50 

11 

.83  527 

.86  283 

.97  244 

.02  756 

.13  717 

.16  473 

49 

12 

.83  540 

.86  271 

.97  269 

.02  731 

.13  729 

.16  460 

48 

13 

.83  554 

.86  259 

.97  295 

.02  705 

.13  741 

.16  446 

47 

14 

.83  567 

.86  247 

.97  320 

.02  680 

.13  753 

.16  433 

46 

15 

9.83  581 

9.86  235 

9.97  345 

0.02  655 

0.13  765 

0.16  419 

45 

16 

.83  594 

.86  223 

.97  371 

.02  629 

.13  777 

.16  406 

44 

17 

.83  608 

.86  211 

.  .97  396 

.02  604 

.13  789 

.16  392 

43 

18 

.83  621 

.86  200 

.97  421 

.02  579 

.13  800 

.16  379 

42 

19 

.83  634 

.86  188 

.97  447 

.02  553 

.13  812 

.16  366 

41 

20 

9.83  648 

9.86  176 

9.97  472 

0.02  528 

0.13  824 

0.16  352 

40 

21 

.83  661 

.86  164 

.97  497 

.02  503 

.13  836 

.16  339 

39 

22 

.83  674 

.86  152 

.97  523 

.02  477 

.13  848 

.16  326 

38 

23 

.83  688 

.86  140 

.97  548 

.02  452 

.13  860 

.16  312 

37 

24 

.83  701 

.86  128 

.97  573 

.02  427 

.13  872 

.16  299 

36 

25 

9.83  715 

9.86  116 

9.97  598 

0.02  402 

0.13  884 

0.16  285 

35 

26 

.83  728 

.86  104 

.97  624 

.02  376 

.13  896 

.16  272 

34 

27 

.83  741 

.86  092 

.97  649 

.02  351 

.13  908 

.16  259 

33 

28 

.83  755 

.86  080 

.97  674 

.02  326 

.13  920 

.16  245 

32 

29 

.83  768 

.86  068 

.97  700 

.02  300 

.13  932 

.16  232 

31 

30 

9.83  781 

9.86  056 

9.97  725 

0.02  275 

0.13  944 

0.16  219 

30 

.  31 

.83  795 

.86  044 

.97  750 

.02  250 

.13  956 

.16  205 

29 

32 

.83  808 

.86  032 

.97  776 

.02  224 

.13  968 

.16  192 

28 

33 

.83  821 

.86  020 

.97  801 

.02  199 

.13  980 

.16  179 

27 

34 

.83  834 

.86  008 

.97  826 

.02  174 

.13  992 

.16  166 

26 

35 

9.83  848 

9.85  996 

9.97  851 

0.02  149 

0.14  004 

0.16  152 

25 

36 

.83  861 

.85  984 

.97  877 

.02  123 

.14  016 

.16  139 

24 

37 

.83  874 

.85  972 

.97  902 

.02  098 

.14  028 

.16  126 

23 

38 

.83  887 

.85  960 

.97  927 

.02  073 

.14  040 

.16  113 

22 

39 

.83  901 

.85  948 

.97  953 

.02  047 

.14  052 

.16  099 

21 

40 

9.83  914 

9.85  936 

9.97  978 

0.02  022 

0.14  064 

0.16  086 

20 

41 

.83  927 

.85  924 

.98  003 

.01  997 

.14  076 

.16  073 

19 

42 

.83  940 

.85  912 

.98  029 

.01  971 

.14  088 

.16  060 

18 

43 

.83  954 

.85  900 

.98  054 

.01  946 

.14  100 

.16  046 

17 

44 

.83  967 

.85  888 

.98  079 

.01  921 

.14  112 

.16  033 

16 

45 

9.83  980 

9.85  876 

9.98  104 

0.01  896 

0.14  124 

0.16  020 

15 

46 

.83  993 

.85  864 

.98  130 

.01  870 

.14  136 

.16  007 

14 

47 

.84  006 

.85  851 

.98  155 

.01  845 

.14  149 

.15  994 

13 

48 

.84  020 

.85  839 

.98  180 

.01  820 

.14  161 

.15  980 

12 

49 

.84  033 

.85  827 

.98  206 

.01  794 

.14  173 

.15  967 

11 

50 

9.84  046 

9.85  815 

9.98  231 

0.01  769 

0.14  185 

0.15  954 

10 

51 

.84  059 

.85  803 

.98  256 

.01  744 

.14  197 

.15  941 

9 

52 

.84  072 

.85  791 

.98  281 

.01  719 

.14  209 

.15  928 

8 

53 

.84  085 

.85  779 

.98  307 

.01  693 

.14  221 

.15  915 

7 

54 

.84  098 

.85  766 

.98  332 

.01  668 

.14  234 

.15  902 

6 

55 

9.84  112 

9.85  754 

9.98  357 

0.01  643 

0.14  246 

0.15  888 

5 

56 

.84  125 

.85  742 

.98  383 

.01  617 

.14  258 

.15  875 

4 

57 

.84  138 

.85  730 

.98  408 

.01  592 

.14  270 

.15  862 

3 

58 

.84  151 

.85  718 

.98  433 

.01  567 

.14  282 

.15  849 

2 

59 

.84  164 

.85  706 

.98  458 

.01  542 

.14  294 

.15  836 

1 

60 

9.84  177 

9.85  693 

9.98  484 

0.01  516 
Tan 

0.14  307 

0.15  823 
Sec 

0 

Cos 

Sin 

Cot 

Csc 

/ 

133°  (313°) 


(226°)  46' 


240 


Table  4.    Trigonometric  Logarithms 


44°  (224°) 

(315°)  135° 

/ 

Sin 

Cos 

Tan 

Cot 

Sec 

Csc 

0 

9.84  177 

9.85  693 

9.98  484 

0.01  516 

0.14  307 

0.15  823 

60 

1 

.84  190 

.85  681 

.98  509 

.01  491 

.14  319 

.15  810 

59 

2 

.84  203 

.85  669 

.98  534 

.01  466 

.14  331 

.15  797 

58 

3 

.84  216 

.85  657 

.98  560 

.01  440 

.14  343 

.15  784 

57 

4 

.84  229 

.85  645 

.98  585 

.01  415 

.14  355 

.15  771 

56 

5 

9.84  242 

9.85  632 

9.98  610 

0.01  390 

0.14  368 

0.15  758 

55 

6 

.84  255 

.85  620 

.98  635 

.01  365 

.14  380 

.15  745 

54 

7 

.84  269 

.85  608 

.98  661 

.01  339 

.14  392 

.15  731 

53 

8 

.84  282 

.85  596 

.98  686 

.01  314 

.14  404 

.15  718 

52 

9 

.84  295 

.85  583 

.98  711 

.01  289 

.14  417 

.15  705 

51 

10 

9.84  308 

9.85  571 

9.98  737 

0.01  263 

0.14  429 

0.15  692 

50 

11 

.84  321 

.85  559 

.98  762 

.01  238 

.14  441 

.15  679 

49 

12 

.84  334 

.85  547 

.98  787 

.01  213 

.14  453 

.15  666 

48 

13 

.84  347 

.85  534 

.98  812 

.01  188 

.14  466 

.15  653 

47 

14 

.84  360 

.85  522 

.98  838 

.01  162 

.14  478 

.15  640 

46 

15 

9.84  373 

9.85  510 

9.98  863 

0.01  137 

0.14  490 

0.15  627 

45 

16 

.84  385 

.85  497 

.98  888 

.01  112 

.14  503 

.15  615 

44 

17 

.84  398 

.85  485 

.98  913 

.01  087 

.14  515 

.15  602 

43 

18 

.84  411 

.85  473 

.98  939 

.01  061 

.14  527 

.15  589 

42 

19 

.84  424 

.85  460 

.98  964 

.01  036 

.14  540 

.15  576 

41 

20 

9.84  437 

9.85  448 

9.98  989 

0.01  Oil 

0.14  552 

0.15  563 

40 

21 

.84  450 

.85  436 

.99  015 

.00  985 

.14  564 

.15  550 

39 

22 

.84  463 

.85  423 

.99  040 

.00  960 

.14  577 

.15  537 

38 

23 

.84  476 

.85  411 

.99  065 

.00  935 

.14  689 

.15  524 

37 

24 

.84  489 

.85  399 

.99  090 

.00  910 

.14  601 

.15  511 

36 

25 

9.84  502 

9.85  386 

9.99  116 

0.00  884 

0.14  614 

0.15  498 

35 

26 

.84  515 

.85  374 

.99  141 

.00  859 

.14  626 

.15  485 

34 

27 

.84  528 

.85  361 

.99  166 

.00  834 

.14  639 

.15  472 

33 

28 

.84  540 

.85  349 

.99  191 

.00  809 

.14  651 

.15  460 

32 

29 

.84  553 

.85  337 

.99  217 

.00  783 

.14  663 

.15  447 

31 

30 

9.84  566 

9.85  324 

9.99  242 

0.00  758 

0.14  676 

0.15  434 

30 

31 

.84  579 

.85  312 

.99  267 

.00  733 

.14  688 

.15  421 

29  . 

32 

.84  592 

.85  299 

.99  293 

.00  707 

.14  701 

.15  408 

28 

33 

.84  605 

.85  287 

.99  318 

.00  682 

.14  713 

.15  395 

27 

34 

.84  618 

.85  274 

.99  343 

.00  657 

.14  726 

.15  382 

26 

35 

9.84  630 

9.85  262 

9.99  368 

0.00  632 

0.14  738 

0.15  370 

25 

36 

.84  643 

.85  250 

.99  394 

.00  606 

.14  750 

.15  357 

24 

37 

.84  656 

.85  237 

.99  419 

.00  581 

.14  763 

.15  344 

23 

38 

.84  669 

.85  225 

.99  444 

.00  556 

.14  775 

.15  331 

22 

39 

.84  682 

.85  212 

.99  469 

.00  531 

.14  788 

.15  318 

21 

40 

9.84  694 

9.85  200 

9.99  495 

0.00  505 

0.14  800 

0.15  306 

20 

41 

.84  707 

.85  187 

.99  520 

.00  480 

.14  813 

.15  293 

19 

42 

.84  720 

.85  175 

.99  545 

.00  455 

.14  825 

.15  280 

18 

43 

.84  733 

.85  162 

.99  570 

.00  430 

.14  838 

.15  267 

17 

44 

.84  745 

.85  150 

.99  596 

.00  404 

.14  850 

.15  255 

16 

45 

9.84  758 

9.85  137 

9.99  621 

0.00  379 

0.14  863 

0.15  242 

15 

46 

.84  771 

.85  125 

.99  646 

.00  354 

.14  875 

.15  229 

14 

47 

.84  784 

.85  112 

.99  672 

.00  328 

.14  888 

.15  216 

13 

48 

.84  796 

.85  100 

.99  697 

.00  303 

.14  900 

.15  204 

12 

49 

.84  809 

.85  087 

.99  722 

.00  278 

.14  913 

.15  191 

11 

50 

9.84  822 

9.85  074 

9.99  747 

0.00  253 

0.14  926 

0.15  178 

10 

51 

.84  835 

.85  062 

.99  773 

.00  227 

.14  938 

.15  165 

9 

52 

.84  847 

.85  049 

.99  798 

.00  202 

.14  951 

.15  153 

8 

53 

.84  860 

.85  037 

.99  823 

.00  177 

.14  963 

.15  140 

7 

54 

.84  873 

.85  024 

.99  848 

.00  152 

.14  976 

.15  127 

6 

55 

9.84  885 

9.85  012 

9.99  874 

0.00  126 

0.14  988 

0.15  115 

5 

56 

.84  898 

.84  999 

.99  899 

.00  101 

.15  001 

.15  102 

4 

57 

.84  911 

.84  986 

.99  924 

.00  076 

.15  014 

.15  089 

3 

58 

.84  923 

.84  974 

.99  949 

.00  051 

.15  026 

.15  077 

2 

59 

.84  936 

.84  961 

.99  975 

.00  025 

.15  039 

.15  064 

1 

60 

9.84  949 

9.84  949 

0.00  000 

0.00  000 

0.15  051 

0.15  051 

0 

Cos 

Sin 

Cot 

Tan 

Csc 

Sec 

134°  (314°) 


(225°)  45^ 


Table  5.    Meridional  Parts 


241 


/ 

0° 

1° 

2° 

3° 

4° 

5° 

6° 

7° 

8° 

9° 

/ 

0 

0.0 

59.6 

119.2 

178.9 

238.6 

298.3 

358.2 

418.2 

478.3 

538.6 

0 

1 

1.0 

60.6 

20.2 

79.9 

39.6 

99.3 

59.2 

19.2 

79.3 

39.6 

1 

2 

2.0 

61.6 

21.2 

80.8 

40.6 

300.3 

60.2 

20.2 

80.3 

40.6 

2 

3 

3.0 

62.6 

22.2 

81.8 

41.6 

01.3 

61.2 

21.2 

81.3 

41.6 

3 

4 

4.0 

63.6 

23.2 

82.8 

42.5 

02.3 

62.2 

22.2 

82.3 

42.6 

4 

5 

5.0 

64.6 

124.2 

183.8 

243.5 

303.3 

363.2 

423.2 

483.3 

543.6 

5 

6 

6.0 

65.6 

25.2 

84.8 

44.5 

04.3 

64.2 

24.2 

84.3 

44.6 

6 

7 

7.0 

66.5 

26.2 

85.8 

45.5 

05.3 

65.2 

25.2 

85.3 

45.6 

7 

8 

7.9 

67.5 

27.2 

86.8 

46.5 

06.3 

66.2 

26.2 

86.3 

46.6 

8 

9 

8.9 

68.5 

28.2 

87.8 

47.5 

07.3 

67.2 

27.2 

87.3 

47.6 

9 

10 

9.9 

69.5 

129.1 

188.8 

248.5 

308.3 

368.2 

428.2 

488.3 

548.6 

10 

11 

10.9 

70.5 

30.1 

89.8 

49.5 

09.3 

69.2 

29.2 

89.3 

49.6 

11 

12 

11.9 

71.5 

31.1 

90.8 

50.5 

10.3 

70.2 

30.2 

90.4 

50.6 

12 

13 

12.9 

72.5 

32.1 

91.8 

51.5 

11.3 

71.2 

31.2 

91.4 

51.7 

13 

14 

13.9 

73.5 

33.1 

92.8 

52.5 

12.3 

72.2 

32.2 

92.4 

52.7 

14 

15 

14.9 

74.5 

134.1 

193.8 

253.5 

313.3 

373.2 

433.2 

493.4 

553.7 

15 

16 

15.9 

75.5 

35.1 

94.8 

54.5 

14.3 

74.2 

34.2 

94.4 

54.7 

16 

17 

16.9 

76.5 

36.1 

95.8 

55.5 

15.3 

75.2 

35.2 

95.4 

55.7 

17 

18 

17.9 

77.5 

37.1 

96.8 

56.5 

16.3 

76.2 

36.2 

96.4 

56.7 

18 

19 

18.9 

78.5 

38.1 

97.8 

57.5 

17.3 

77.2 

37.2 

97.4 

57.7 

19 

20 

19.9 

79.5 

139.1 

198.8 

258.5 

318.3 

378.2 

438.2 

498.4 

558.7 

20 

21 

20.9 

80.5 

40.1 

99.7 

59.5 

19.3 

79.2 

39.2 

99.4 

59.7 

21 

22 

21.9 

81.5 

41.1 

200.7 

60.5 

20.3 

80.2 

40.2 

500.4 

60.7 

22 

23 

22.8 

82.4 

42.1 

01.7 

61.5 

21.3 

81.2 

41.2 

01.4 

61.7 

23 

24 

23.8 

83.4 

43.1 

02.7 

62.5 

22.3 

82.2 

42.2 

02.4 

62.7 

24 

25 

24.8 

84.4 

144.1 

203.7 

263.5 

323.3 

383.2 

443.2 

503.4 

563.7 

25 

26 

25.8 

85.4 

45.1 

04.7 

64.5 

24.3 

84.2 

44.2 

04.4 

64.7 

26 

27 

26.8 

86.4 

46.0 

05.7 

65.5 

25.3 

85.2 

45.2 

05.4 

65.7 

27 

28 

27.8 

87.4 

47.0 

06.7 

66.5 

26.3 

86.2 

46.2 

06.4 

66.8 

28 

29 

28.8 

88.4 

48.0 

07.7 

67.4 

27.3 

87.2 

47.2 

07.4 

67.8 

29 

30 

29.8 

89.4 

149.0 

208.7 

268.4 

328.3 

388.2 

448.2 

508.4 

568.8 

30 

31 

30.8 

90.4 

50.0 

09.7 

69.4 

29.3 

89.2 

49.2 

09.4 

69.8 

31 

32 

31.8 

91.4 

51.0 

10.7 

70.4 

30.3 

90.2 

50.2 

10.4 

70.8 

32 

33 

32.8 

92.4 

52.0 

11.7 

71.4 

31.3 

91.2 

51.2 

11.4 

71.8 

33 

34 

33.8 

93.4 

53.0 

12.7 

72.4 

32.3 

92.2 

52.2 

12.4 

72.8 

34 

35 

34.8 

94.4 

154.0 

213.7 

273.4 

333.3 

393.2 

453.2 

513.4 

573.8 

35 

36 

35.8 

95.4 

55.0 

14.7 

74.4 

34.3 

94.2 

54.3 

14.5 

74.8 

36 

37 

36.7 

96.4 

56.0 

15.7 

75.4 

35.3 

95.2 

55.3 

15.5 

75.8 

37 

38 

37.7 

97.3 

57.0 

16.7 

76.4 

36.2 

96.2 

56.3 

16.5 

76.8 

38 

39 

38.7 

98.3 

58.0 

17.7 

77.4 

37.2 

97.2 

57.3 

17.5 

77.8 

39 

40 

39.7 

99.3 

159.0 

218.7 

278.4 

338.2 

398.2 

458.3 

518.5 

578.8 

40 

41 

40.7 

100.3 

60.0 

19.7 

79.4 

39.2 

99.2 

59.3 

19.5 

79.9 

41 

42 

41.7 

01.3 

61.0 

20.6 

80.4 

40.2 

400.2 

60.3 

20.5 

80.9 

42 

43 

42.7 

02.3 

62.0 

21.6 

81.4 

41.2 

01.2 

61.3 

21.5 

81.9 

43 

44 

43.7 

03.3 

63.0 

22.6 

82.4 

42.2 

02.2 

62.3 

22.5 

82.9 

44 

45 

44.7 

104.3 

164.0 

223.6 

283.4 

343.2 

403.2 

463.3 

523.5 

583.9 

45 

46 

45.7 

05.3 

65.0 

24.6 

84.4 

44.2 

04.2 

64.3 

24.5 

84.9 

46 

47 

46.7 

06.3 

66.0 

25.6 

85.4 

45.2 

05.2 

65.3 

25.5 

85.9 

47 

48 

47.7 

07.3 

67.0 

26.6 

86.4 

46.2 

06.2 

66.3 

26.5 

86.9 

48 

49 

48.7 

08.3 

68.0 

27.6 

87.4 

47.2 

07.2 

67.3 

27.5 

87.9 

49 

50 

49.7 

109.3 

168.9 

228.6 

288.4 

348.2 

408.2 

468.3 

528.5 

588.9 

50 

51 

50.7 

10.3 

69.9 

29.6 

89.4 

49.2 

09.2 

69.3 

29.5 

89.9 

51 

52 

51.6 

11.3 

70.9 

30.6 

90.4 

50.2 

10.2 

70.3 

30.5 

90.9 

52 

53 

52.6 

12.3 

71.9 

31.6 

91.4 

51.2 

11.2 

71.3 

31.5 

91.9 

53 

54 

53.6 

13.2 

72.9 

32.6 

92.4 

52.2 

12.2 

72.3 

32.5 

93.0 

54 

55 

54.6 

114.2 

173.9 

233.6 

293.4 

353.2 

413.2 

473.3 

533.5 

594.0 

55 

56 

55.6 

15.2 

74.9 

34.6 

94.4 

54.2 

14.2 

74.3 

34.6 

95.0 

56 

67 

56.6 

16.2 

75.9 

35.6 

95.4 

55.2 

15.2 

75.3 

35.6 

96.0 

57 

58 

57.6 

17.2 

76.9 

36.6 

96.3 

56.2 

16.2 

76.3 

36.6 

97.0 

58 

59 

58.6 

18.2 

77.9 

37.6 

97.3 

57.2 

17.2 

77.3 

37.6 

98.0 

59 

60 

59.6 

119.2 

178.9 

238.6 

298.3 

358.2 

418.2 

478.3 

538.6 

599.0 

60 

/ 

0° 

1° 

2° 

3° 

4° 

5° 

6° 

7° 

8° 

9° 

/ 

242 


Table  5.    Meridional  Parts 


t 

10° 

11° 

12° 

13° 

14° 

15° 

16° 

17° 

18° 

19° 

/ 

0 

599.0 

659.6 

720.5 

781.5 

842.8 

904.4 

966.3 

1028.5 

1091.0 

1153.9 

0 

1 

600.0 

60.6 

21.5 

82.5 

43.9 

05.4 

67.3 

29.5 

92.0 

54.9 

1 

2 

01.0 

61.7 

22.5 

83.6 

44.9 

06.5 

68.3 

30.5 

93.1 

56.0 

2 

3 

02.0 

62.7 

23.5 

84.6 

45.9 

07.5 

69.4 

31.6 

94.1 

57.0 

3 

4 

03.0 

63.7 

24.5 

85.6 

46.9 

08.5 

70.4 

32.6 

95.2 

58.1 

4 

5 

604.1 

664.7 

725.5 

786.6 

847.9 

909.6 

971.4 

1033.7 

1096.2 

1159.1 

5 

6 

05.1 

65.7 

26.6 

87.6 

49.0 

10.6 

72.5 

34.7 

97.3 

60.2 

6 

7 

06.1 

66.7 

27.6 

88.7 

50.0 

11.6 

73.5 

35.7 

98.3 

61.2 

7 

8 

07.1 

67.7 

28.6 

89.7 

51.0 

12.6 

74.6 

36.8 

99.4 

62.3 

8 

9 

08.1 

68.7 

29.6 

90.7 

52.0 

13.7 

75.6 

37.8 

1100.4 

63.3 

9 

10 

609.1 

669.8 

730.6 

791.7 

853.1 

914.7 

976.6 

1038.9 

1101.4 

1164.4 

10 

11 

10.1 

70.8 

31.6 

92.7 

54.1 

15.7 

77.7 

39.9 

02.5 

65.4 

11 

12 

11.1 

71.8 

32.7 

93.8 

55.1 

16.8 

78.7 

40.9 

03.5 

66.5 

12 

13 

12.1 

72.8 

33.7 

94.8 

56.1 

17.8 

79.7 

42.0 

04.6 

67.5 

13 

14 

13.1 

73.8 

34.7 

95.8 

57.2 

18.8 

80.8 

43.0 

05.6 

68.6 

14 

15 

614.1 

674.8 

735.7 

796.8 

858.2 

919.8 

981.8 

1044.1 

1106.7 

1169.7 

15 

16 

15.2 

75.8 

36.7 

97.8 

59.2 

20.9 

82.8 

45.1 

07.7 

70.7 

16 

17 

16.2 

76.8 

37.7 

98.9 

60.2 

21.9 

83.9 

46.1 

08.8 

71.8 

17 

18 

17.2 

77.9 

38.8 

99.9 

61.3 

22.9 

84.9 

47.2 

09.8 

72.8 

18 

19 

18.2 

78.9 

39.8 

800.9 

62.3 

24.0 

85.9 

48.2 

10.9 

73.9 

19 

20 

619.2 

679.9 

740.8 

801.9 

863.3 

925.0 

987.0 

1049.3 

1111.9 

1174.9 

20 

21 

20.2 

80.9 

41.8 

02.9 

64.3 

26.0 

88.0 

50.3 

13.0 

76.0 

21 

22 

21.2 

81.9 

42.8 

04.0 

65.4 

27.1 

89.0 

51.3 

14.0 

77.0 

22 

23 

22.2 

82.9 

43.8 

05.0 

66.4 

28.1 

90.1 

52.4 

15.0 

78.1 

23 

24 

23.2 

83.9 

44.9 

06.0 

67.4 

29.1 

91.1 

53.4 

16.1 

79.1 

24 

25 

624.2 

684.9 

745.9 

807.0 

868.5 

930.1 

992.1 

1054.5 

1117.1 

1180.2 

25 

26 

25.3 

86.0 

46.9 

.08.1 

69.5 

31.2 

93.2 

55.5 

18.2 

81.2 

26 

27 

26.3 

87.0 

47.9 

09.1 

70.5 

32.2 

94.2 

56.6 

19.2 

82.3 

27 

28 

27.3 

88.0 

48.9 

10.1 

71.5 

33.2 

95.3 

57.6 

20.3 

83.3 

28 

29 

28.3 

89.0 

49.9 

11.1 

72.6 

34.3 

96.3 

58.6 

21.3 

84.4 

29 

30 

629.3 

690.0 

751.0 

812.1 

873.6 

935.3 

997.3 

1059.7 

1122.4 

1185.5 

30 

31 

30.3 

91.0 

52.0 

13.2 

74.6 

36.3 

98.4 

60.7 

23.4 

86.5 

31 

32 

31.3 

92.0 

53.0 

14.2 

75.6 

37.4 

99.4 

61.8 

24.5 

87.6 

32 

33 

32.3 

93.1 

54.0 

15.2 

76.7 

38.4 

1000.4 

62.8 

25.5 

88.6 

33 

34 

33.3 

94.1 

55.0 

16.2 

77.7 

39.4 

01.5 

63.9 

26.6 

89.7 

34 

35 

634.3 

695.1 

756.0 

817.3 

878.7 

940.5 

1002.5 

1064.9 

1127.6 

1190.7 

35 

36 

35.4 

96.1 

57.1 

18.3 

79.7 

41.5 

03.6 

65.9 

28.7 

91.8 

36 

37 

36.4 

97.1 

58.1 

19.3 

80.8 

42.5 

04.6 

67.0 

29.7 

92.8 

37 

38 

37.4 

98.1 

59.1 

20.3 

81.8 

43.6 

05.6 

68.0 

30.8 

93.9 

38 

39 

38.4 

99.1 

60.1 

21.3 

82.8 

44.6 

06.7 

69.1 

31.8 

95.0 

39 

40 

639.4 

700.2 

761.1 

822.4 

883.8 

945.6 

1007.7 

1070.1 

1132.9 

1196.0 

40 

41 

40.4 

01.2 

62.2 

23.4 

.84.9 

46.7 

08.7 

71.2 

33.9 

97.1 

41 

42 

41.4 

02.2 

63.2 

24.4 

85.9 

47.7 

09.8 

72.2 

35.0 

98.1 

42 

43 

42.4 

03.2 

64.2 

25.4 

86.9 

48.7 

10.8 

73.2 

36.0 

99.2 

43 

44 

43.4 

04.2 

65.2 

26.5 

88.0 

49.7 

11.8 

74.3 

37.1 

1200.2 

44 

45 

644.5 

705.2 

766.2 

827.5 

889.0 

950.8 

1012.9 

1075.3 

1138.1 

1201.3 

45 

46 

45.5 

06.2 

67.3 

28.5 

90.0 

51.8 

13.9 

76.4 

39.2 

02.3 

46 

47 

46.5 

07.3 

68.3 

29.5 

91.0 

52.8 

15.0 

77.4 

40.2 

03.4 

47 

48 

47.5 

08.3 

69.3 

30.5 

92.1 

53.9 

16.0 

78.5 

41.3 

04.5 

48 

49 

48.5 

09.3 

70.3 

31.6 

93.1 

54.9 

17.0 

79.5 

42.3 

05.5 

49 

50 

649.5 

710.3 

771.3 

832.6 

894.1 

955.9 

1018.1 

1080.5 

1143.4 

1206.6 

50 

51 

50.5 

11.3 

72.3 

33.6 

95.2 

57.0 

19.1 

81.6 

44.4 

07.6 

51 

52 

51.5 

12.3 

73.4 

34.6 

96.2 

58.0 

20.2 

82.6 

45.5 

08.7 

52 

53 

52.5 

13.4 

74.4 

35.7 

97.2 

59.0 

21.2 

83.7 

46.5 

09.7 

53 

54 

53.6 

14.4 

75.4 

36.7 

98.2 

60.1 

22.2 

84.7 

47.6 

10.8 

54 

55 

654.6 

715.4 

776.4 

837.7 

899.3 

961.1 

1023.3 

1085.8 

1148.6 

1211.8 

55 

56 

55.6 

16.4 

77.4 

38.7 

900.3 

62.1 

24.3 

86.8 

49.7 

12.9 

56 

57 

56.6 

17.4 

78.5 

39.8 

01.3 

63.2 

25.3 

87.9 

50.7 

14.0 

57 

58 

57.6 

18.4 

79.5 

40.8 

02.3 

64.2 

26.4 

88.9 

51.8 

15.0 

58 

59 

58.6 

19.4 

80.5 

41.8 

03.4 

65.2 

27.4 

89.9 

52.8 

16.1 

59 

60 

659.6 

720.5 

781.5 

842.8 

904.4 

966.3 

1028.5 

1091.0 

1153.9 

1217.1 

60 

/ 

10° 

11° 

12° 

13° 

14° 

15° 

16° 

17° 

18° 

19° 

/ 

Table  5.    Meridional  Parts 


243 


/ 

20° 

21° 

22° 

23° 

24° 

25° 

26°  1  27° 

28° 

29° 

' 

0 

1217.1 

1280.8 

1344.9 

1409.5 

1474.5 

1540.1 

1606.2!l672.9|l740.2 

1808.1 

0 

1 

18.2 

81.9 

46.0 

10.6 

75.6 

41.2 

07.3 

74.0 

41.3 

09.2 

1 

2 

19.3 

82.9 

47.1 

11.6 

76.7 

42.3 

08.4 

75.1 

42.4 

10.4 

2 

3 

20.3 

84.0 

48.1 

12.7 

77.8 

43.4 

09.5 

76.2 

43.6 

11.5 

3 

4 

21.4 

85.1 

49.2 

13.8 

78.9 

44.5 

10.6 

77.4 

44.7 

12.6 

4 

5 

1222.4 

1286.1 

1350.3 

1414.9 

1480.0 

1545.6 

1611.7 

1678.5 

1745.8 

1813.8 

5 

6 

23.5 

87.2 

51.4 

16.0 

81.1 

46.7 

12.9 

79.6 

46.9 

14.9 

6 

7 

24.5 

88.3 

52.4 

17.1 

82.2 

47.8 

14.0 

80.7 

48.1 

16.1 

7 

8 

25.6 

89.3 

53.5 

18.1 

83.3 

48.9 

15.1 

81.8 

49.2 

17.2 

8 

9 

26.7 

90.4 

54.6 

19.2 

84.3 

50.0 

16.2 

82.9 

50.3 

18.3 

9 

10 

1227.7 

1291.5 

1355.7 

1420.3 

1485.4 

1551.1 

1617.3 

1684.1 

1751.5 

1819.5 

10 

11 

28.8 

92.5 

56.7 

21.4 

80.5 

52.2 

18.4 

85.2 

52.6 

20.6 

11 

12 

29.8 

93.6 

57.8 

22.5 

87.6 

53.3 

19.5 

86.3 

53.7 

21.8 

12 

13 

30.9 

94.7 

58.9 

23.5 

88.7 

54.4 

20.6 

87.4 

54.8 

22.9 

13 

14 

32.0 

95.7 

59.9 

24.6 

89.8 

55.5 

21.7 

88.5 

56.0 

24.0 

14 

15 

1233.0 

1296.8 

1361.0 

1425.7 

1490.9 

1556.6 

1622.8 

1689.7 

1757.1 

1825.2 

15 

16 

34.1 

97.9 

62.1 

26.8 

92.0 

57.7 

23.9 

90.8 

58.2 

26.3 

16 

17 

35.1 

98.9 

63.2 

27.9 

93.1 

58.8 

25.0 

91.9 

59.4 

27.5 

17 

18 

36.2 

1300.0 

64.2 

29.0 

94.2 

59.9 

26.2 

93.0 

60.5 

28.6 

18 

19 

37.3 

01.1 

65.3 

30.0 

95.2 

61.0 

27.3 

94.1 

61.6 

29.7 

19 

20 

1238.3 

1302.1 

1366.4 

1431.1 

1496.3 

1562.1 

1628.4 

1695.3 

1762.7 

1830.9 

20 

21 

39.4 

03.2 

67.5 

32.2 

97.4 

63.2 

29.5 

96.4 

63.9 

32.0 

21 

22 

40.4 

04.3 

68.5 

33.3 

98.5 

64.3 

30.6 

97.5 

65.0 

33.2 

22 

23 

41.5 

05.3 

69.6 

34.4 

99.6 

65.4 

31.7 

98.6 

66.1 

34.3 

23 

24 

42.6 

06.4 

70.7 

35.4 

1500.7 

66.5 

32.8 

99.7 

67.3 

35.4 

24 

25 

1243.6 

1307.5 

1371.8 

1436.5 

1501.8 

1567.6 

1633.9 

1700.9 

1768.4 

1836.6 

25 

26 

44.7 

08.5 

72.8 

37.6 

02.9 

68.7 

35.0 

02.0 

69.5 

37.7 

26 

27 

45.7 

09.6 

73.9 

38.7 

04.0 

69.8 

36.1 

03.1 

70.7 

38.9 

27 

28 

46.8 

10.7 

75.0 

39.8 

05.1 

70.9 

37.3 

04.2 

71.8 

40.0 

28 

29 

47.9 

11.7 

76.1 

40.9 

06.2 

72.0 

38.4 

05.3 

72.9 

41.2 

29 

30 

1248.9 

1312.8 

1377.1 

1442.0 

1507.3 

1573.1 

1639.5 

1706.5 

1774.1 

1842.3 

30 

31 

50.0 

13.9 

78.2 

43.0 

08.4 

74.2 

40.6 

07.6 

75.2 

43.4 

31 

32 

51.0 

14.9 

79.3 

44.1 

09.4 

75.3 

41.7 

08.7 

76.3 

44.6 

32 

33 

52.1 

16.0 

80.4 

45.2 

10.5 

76.4 

42.8 

09.8 

77.4 

45.7 

33 

34 

53.2 

17.1 

81.5 

46.3 

11.6 

77.5 

43.9 

10.9 

78.6 

46.9 

34 

35 

1254.2 

1318.2 

1382.5 

1447.4 

1512.7 

1578.6 

1645.0 

1712.1 

1779.7 

1848.0 

35 

36 

55.3 

19.2 

83.6 

48.5 

13.8 

79.7 

46.2 

13.2 

80.8 

49.2 

36 

37 

56.4 

20.3 

84.7 

49.5 

14.9 

80.8 

47.3 

14.3 

82.0 

50.3 

37 

38 

57.4 

21.4 

85.8 

50.6 

16.0 

81.9 

48.4 

15.4 

83.1 

51.4 

38 

39 

58.5 

22.4 

86.8 

51.7 

17.1 

83.0 

49.5 

16.6 

84.2 

52.6 

39 

40 

1259.5 

1323.5 

1387.9 

1452.8 

1518.2 

1584.1 

1650.6 

1717.7 

1785.4 

1853.7 

40 

41 

60.6 

24.6 

89.0 

53.9 

19.3 

85.2 

51.7 

18.8 

86.5 

54.9 

41 

42 

61.7 

25.6 

90.1 

55.0 

20.4 

86.3 

52.8 

19.9 

87.6 

56.0 

42 

43 

62.7 

26.7 

91.1 

56.1 

21.5 

87.4 

53.9 

21.1 

88.8 

57.2 

43 

44 

63.8 

27.8 

92.2 

57.1 

22.6 

88.5 

55.1 

22.2 

89.9 

58.3 

44 

45 

1264.9 

1328.9 

1393.3 

1458.2 

1523.7 

1589.6 

1656.2 

1723.3 

1791.1 

1859.5 

45 

46 

65.9 

29.9 

94.4 

59.3 

24.8 

90.7 

57.3 

24.4 

92.2 

60.6 

46 

47 

67.0 

31.0 

95.5 

60.4 

25.9 

91.8 

58.4 

25.5 

93.3 

61.8 

47 

48 

68.0 

32.1 

96.5 

61.5 

27.0 

92.9 

59.5 

26.7 

94.5 

62.9 

48 

49 

69.1 

33.1 

97.6 

62.6 

28.0 

94.1 

60.6 

27.8 

95.6 

64.0 

49 

50 

1270.2 

1334.2 

1398.7 

1463.7 

1529.1 

1595.2 

1661.7 

1728.9 

1796.7 

1865.2 

50 

51 

71.2 

35.3 

99.8 

64.8 

30.2 

96.3 

62.9 

30.0 

97.9 

66.3 

51 

52 

72.3 

36.3 

1400.9 

65.8 

31.3 

97.4 

64.0 

31.2 

99.0 

67.5 

52 

53 

73.4 

37.4 

01.9 

66.9 

32.4 

98.5 

65.1 

32.3 

1800.1 

68.6 

53 

54 

74.4 

38.5 

03.0 

68.0 

33.5 

99.6 

66.2 

33.4 

01.3 

69.8 

54 

55 

1275.5 

1339.6 

1404.1 

1469.1 

1534.6 

1600.7 

1667.3 

1734.^ 

1802.4 

1870.9 

55 

56 

76.6 

40.6 

05.2 

70.2 

35.7 

01.8 

68.4 

35.7 

03.5 

72.1 

56 

57 

77.6 

41.7 

06.2 

71.3 

36.8 

02.9 

69.5 

36.8 

04.7 

73.2 

57 

58 

78.7 

42.8 

07.3 

72.4 

37.9 

04.0 

70.7 

37,9 

05.8 

74.4 

58 

59 

79.7 

43.8 

08.4 

73.5 

39.0 

05.1 

71.8 

39.1 

07.0 

75.5 

59 

60 

1280.8 

1344.9 

1409.5 

1474.5 

1540.1 

1606.2 

1672.9 

1740.2 

1808.1 

1876.7 

60 

t 

20° 

21° 

22° 

23° 

24° 

25° 

26° 

27° 

28° 

29° 

244 


Table  5.    Meridional  Parts 


/ 

30° 

31° 

32° 

33° 

34° 

35° 

36° 

2304.2 

37° 

38° 

39° 

/ 

0 

1876.7 

1946.0 

2016.0 

2086.8 

2158.4 

2230.9 

2378.5 

2453.8 

2530.2 

0 

1 

77.8 

47.1 

17.2 

88.0 

59.6 

32.1 

05.5 

79.8 

55.1 

31.5 

1 

2 

79.0 

48.3 

18.3 

89.2 

60.8 

33.3 

06.7 

81.0 

56.4 

32.8 

2 

3 

80.1 

49.4 

19.5 

90.3 

62.0 

34.5 

07.9 

82.3 

57.6 

34.0 

3 

4 

81.3 

50.6 

20.7 

91.5 

63.2 

35.7 

09.2 

83.5 

58.9 

35.3 

4 

5 

1882.4 

1951.8 

2021.9 

2092.7 

2164.4 

2236.9 

2310.4 

2384.8 

2460.2 

2536.6 

5 

6 

83.6 

52.9 

23.0 

93.9 

65.6 

38.2 

11.6 

86.0 

61.4 

37.9 

6 

7 

84.7 

54.1 

24.2 

95.1 

66.8 

39.4 

12.9 

87.3 

62.7 

39.2 

7 

8 

85.9 

55.3 

25.4 

96.3 

68.0 

40.6 

14.1 

88.5 

64.0 

40.5 

8 

9 

87.0 

56.4 

26.6 

97.5 

69.2 

41.8 

15.3 

89.8 

65.2 

41.7 

9 

10 

1888.2 

1957.6 

2027.7 

2098.7 

2170.4 

2243.0 

2316.5 

2391.0 

2466.5 

2543.0 

10 

11 

89.3 

58.7 

28.9 

99.8 

71.6 

44.2 

17.8 

92.3 

67.8 

44.3 

11 

12 

90.5 

59.9 

30.1 

2101.0 

72.8 

45.5 

19.0 

93.5 

69.0 

45.6 

12 

13 

91.6 

61.1 

31.3 

02.2 

74.0 

46.7 

20.3 

94.8 

70.3 

46.9 

13 

14 

92.8 

62.2 

32.4 

03.4 

75.2 

47.9 

21.5 

96.0 

71.6 

48.2 

14 

15 

1893.9 

1963.4 

2033.6 

2104.6 

2176.4 

2249.1 

2322.7 

2397.3 

2472.8 

2549.5 

15 

16 

95.1 

64.6 

34.8 

05.8 

77.6 

50.3 

24.0 

98.5 

74.1 

50.7 

16 

17 

96.2 

65.7 

36.0 

07.0 

78.8 

51.6 

25.2 

99.8 

75.4 

52.0 

17 

18 

97.4 

66.9 

37.1 

08.2 

80.0 

52.8 

26.4 

2401.0 

76.6 

53.3 

18 

19 

98.5 

68.1 

38.3 

09.4 

81.2 

54.0 

27.7 

02.3 

77.9 

54.6 

19 

20 

1899.7 

1969.2 

2039.5 

2110.6 

2182.5 

2255.2 

2328.9 

2403.5 

2479.2 

2555.9 

20 

21 

1900.8 

70.4 

40.7 

11.8 

83.7 

56.4 

30.1 

04.8 

80.4 

57.2 

21 

22 

02.0 

71.5 

41.8 

12.9 

84.9 

57.7 

31.4 

06.0 

81.7 

58.5 

22 

23 

03.1 

72.7 

43.0 

14.1 

86.1 

58.9 

32.6 

07.3 

83.0 

59.8 

23 

24 

04.3 

73.9 

44.2 

15.3 

87.3 

60.1 

33.8 

08.5 

84.3 

61.0 

24 

25 

1905.5 

1975.0 

2045.4 

2116.5 

2188.5 

2261.3 

2335.1 

2409.8 

2485.5 

2562.3 

25 

26 

06.6 

76.2 

46.6 

17.7 

89.7 

62.5 

36.3 

11.1 

86.8 

63.6 

26 

27 

07.8 

77.4 

47.7 

18.9 

90.9 

63.8 

37.6 

12.3 

88.1 

64.9 

27 

28 

08.9 

78.5 

48.9 

20.1 

92.1 

65.0 

38.8 

13.6 

89.3 

66.2 

28 

29 

10.1 

79.7 

50.1 

21.3 

93.3 

66.2 

40.0 

14.8 

90.6 

67.5 

29 

30 

1911.2 

1980.9 

2051.3 

2122.5 

2194.5 

2267.4 

2341.3 

2416.1 

2491.9 

2568.8 

30 

31 

12.4 

82.0 

52.5 

23.7 

95.7 

68.7 

42.5 

17.3 

93.2 

70.1 

31 

32 

13.5 

83.2 

53.6 

24.9 

96.9 

69.9 

43.7 

18.6 

94.4 

71.4 

32 

33 

14.7 

84.4 

54.8 

26.1 

98.1 

71.1 

45.0 

19.8 

95.7 

72.7 

33 

34 

15.8 

85.5 

56.0 

27.3 

99.4 

72.3 

46.2 

21.1 

97.0 

73.9 

34 

35 

1917.0 

1986.7 

2057.2 

2128.5 

2200.6 

2273.5 

2347.5 

2422.3 

2498.3 

2575.2 

35 

36 

18.2 

87.9 

58.4 

29.6 

01.8 

74.8 

48.7 

23.6 

99.5 

76.5 

36 

37 

19.3 

89.1 

59.5 

30.8 

03.0 

76.0 

49.9 

24.9 

2500.8 

77.8 

37 

38 

20.5 

90.2 

60.7 

32.0 

04.2 

77.2 

51.2 

26.1 

02.1 

79.1 

38 

39 

21.6 

91.4 

61.9 

33.2 

05.4 

78.4 

52.4 

27.4 

03.4 

80.4 

39 

40 

1922.8 

1992.6 

2063.1 

2134.4 

2206.6 

2279.7 

2353.7 

2428.6 

2504.6 

2581.7 

40 

41 

23.9 

93.7 

64.3 

35.6 

07.8 

80.9 

54.9 

29.9 

05.9 

83.0 

41 

42 

25.1 

94.9 

65.5 

36.8 

09.0 

82.1 

56.1 

31.2 

07.2 

84.3 

42 

43 

26.3 

96.1 

66.6 

38.0 

10.2 

83.3 

57.4 

32.4 

08.5 

85.6 

43 

44 

27.4 

97.2 

67.8 

39.2 

11.5 

84.6 

58.6 

33.7 

09.7 

86.9 

44 

45 

1928.6 

1998.4 

2069.0 

2140.4 

2212.7 

2285.8 

2359.9 

2434.9 

2511.0 

2588.2 

45 

46 

29.7 

99.6 

70.2 

41.6 

13.9 

87.0 

61.1 

36.2 

12.3 

89.5 

46 

47 

30.9 

2000.7 

71.4 

42.8 

15.1 

88.3 

62.4 

37.4 

13.6 

90.8 

47 

48 

32.0 

01.9 

72.6 

44.0 

16.3 

89.5 

63.6 

38.7 

14.8 

92.1 

48 

49 

33.2 

03.1 

73.7 

45.2 

17.5 

90.7 

64.8 

40.0 

16.1 

93.4 

49 

50 

1934.4 

2004.3 

2074.9 

2146.4 

2218.7 

2291.9 

2366.1 

2441.2 

2517.4 

2594.7 

50 

51 

35.5 

05.4 

76.1 

47.6 

19.9 

93.2 

•67.3 

42.5 

18.7 

96.0 

51 

52 

36.7 

06.6 

77.3 

48.8 

21.1 

94.4 

68.6 

43.7 

20.0 

97.3 

52 

53 

37.8 

07.8 

78.5 

50.0 

22.4 

95.6 

69.8 

45.0 

21.2 

98.5 

53 

54 

39.0 

08.9 

79.7 

51.2 

23.6 

96.9 

71.1 

46.3 

22.5 

99.8 

54 

55 

1940.2 

2010.1 

2080.8 

2152.4 

2224.8 

2298.1 

2372.3 

2447.5 

2523.8 

2601.1 

55 

56 

41.3 

11.3 

82.0 

53.6 

26.0 

99.3 

73.6 

48.8 

25.1 

02.4 

56 

57 

42.5 

12.5 

83.2 

54.8 

27.2 

2300.5 

74.8 

50.1 

26.4 

03.7 

57 

58 

43.6 

13.6 

84.4 

56.0 

28.4 

01.8 

76.1 

51.3 

27.6 

05.0 

58 

59 

44.8 

14.8 

85.6 

57.2 

29.6 

03.0 

77.3 

52.6 

28.9 

06.3 

59 

60 

1946.0 

2016.0 

2086.8 

2158.4 

2230.9 

2304.2 

2378.5 

2453.8 

2530.2 

2607.6 

60 

/ 

30° 

31° 

32° 

33° 

34° 

35° 

36° 

37° 

38° 

39° 

Table  5.    Meridional  Parts 


245 


/ 

40° 

41° 

42° 

43° 

44° 

45° 

46°   47°  1 

48° 

49° 

3364.4 

1 

0 

2607.6 

2686.2 

2766.0 

2847.1 

2929.5 

3013.4 

3098.7 

3185.6 

3274.1 

0 

1 

08.9 

87.6 

67.4 

48.5 

30.9 

14.8 

3100.1 

87.1 

75.6 

65.9 

1 

2 

10.2 

88.9 

68.7 

49.9 

32.3 

16.2 

01.6 

88.5 

77.1 

67.4 

2 

3 

11.5 

90.2 

70.1 

51.2 

33.7 

17.6 

03.0 

90.0 

78.6 

69.0 

3 

4 

12.8 

91.5 

71.4 

52.6 

35.1 

19.0 

04.4 

91.4 

80.1 

70.5 

4 

5 

2614.1 

2692.8 

2772.8 

2853.9 

2936.5 

3020.4 

3105.9 

3192.9 

3281.6 

3372.0 

5 

6 

15.4 

94.2 

74.1 

55.3 

37.9 

21.8 

07.3 

94.4 

83.1 

73.5 

6 

7 

16.8 

95.5 

75.4 

56.7 

39.3 

23.3 

08.8 

95.8 

84.6 

75.1 

7 

8 

18.1 

96.8 

76.8 

58.0 

40.6 

24.7 

10.2 

97.3 

86.1 

76.6 

8 

9 

19.4 

98.1 

78.1 

59.4 

42.0 

26.1 

11.6 

98.8 

87.6 

78.1 

9 

10 

2620.7 

2699.5 

2779.5 

2860.8 

2943.4 

3027.5 

3113.1 

3200.2 

3289.0 

3379.6 

10 

11 

22.0 

2700.8 

80.8 

62.1 

44.8 

28.9 

14.5 

01.7 

90.5 

81.2 

11 

12 

23.3 

02.1 

82.2 

63.5 

46.2 

30.3 

16.0 

03.2 

92.0 

82.7 

12 

13 

24.6 

03.4 

83.5 

64.9 

47.6 

31.7 

17.4 

04.6 

93.5 

84.2 

13 

14 

25.9 

04.8 

84.8 

66.2 

49.0 

33.2 

18.8 

06.1 

95.0 

85.7 

14 

15 

2627.2 

2706.1 

2786.2 

2867.6 

2950.4 

3034.6 

3120.3 

3207.6 

3296.5 

3387.3 

15 

16 

28.5 

07.4 

87.5 

69.0 

51.8 

36.0 

21.7 

09.0 

98.0 

88.8 

16 

17 

29.8 

08.7 

88.9 

70.3 

53.2 

37.4 

23.2 

10.5 

99.5 

90.3 

17 

18 

31.1 

10.1 

90.2 

71.7 

54.5 

38.8 

24.6 

12.0 

3301.0 

91.8 

18 

19 

32.4 

11.4 

91.6 

73.1 

55.9 

40.2 

26.0 

13.4 

02.5 

93.4 

19 

20 

2633.7 

2712.7 

2792.9 

2874.4 

2957.3 

3041.7 

3127.5 

3214.9 

3304.0 

3394.9 

20 

21 

35.0 

14.0 

94.3 

75.8 

58.7 

43.1 

28.9 

16.4 

05.5 

96.4 

21 

22 

36.3 

15.4 

95.6 

77.2 

60.1 

44.5 

30.4 

17.9 

07.0 

98.0 

22 

23 

37.6 

16.7 

97.0 

78.6 

61.5 

45.9 

31.8 

19.3 

08.5 

99.5 

23 

24 

38.9 

18.0 

98.3 

79.9 

62.9 

47.3 

33.3 

20.8 

10.0 

3401.0 

24 

25 

2640.2 

2719.3 

2799.7 

2881.3 

2964.3 

3048.7 

3134.7 

3222.3 

3311.5 

3402.6 

25 

26 

41.6 

20.7 

2801.0 

82.7 

65.7 

50.2 

36.2 

23.7 

13.0 

04.1 

26 

27 

42.9 

22.0 

02.4 

84.0 

67.1 

51.6 

37.6 

25.2 

14.5 

05.6 

27 

28 

44.2 

23.3 

03.7 

85.4 

68.5 

53.0 

39.0 

26.7 

16.0 

07.2 

28 

29 

45.5 

24.7 

05.1 

86.8 

69.9 

54.4 

40.5 

28.2 

17.5 

08.7 

29 

30 

2646.8 

2726.0 

2806.4 

2888.2 

2971.3 

3055.9 

3141.9 

3229.6 

3319.0 

3410.2 

30 

31 

48.1 

27.3 

07.8 

89.5 

72.7 

57.3 

43.4 

31.1 

20.5 

11.8 

31 

32 

49.4 

28.6 

09.1 

90.9 

74.1 

58.7 

44.8 

32.6 

22.1 

13.3 

32 

33 

50.7 

30.0 

10.5 

92.3 

75.5 

60.1 

46.3 

34.1 

23.6 

14.8 

33 

34 

52.0 

31.3 

11.8 

93.7 

76.9 

61.5 

47.7 

35.6 

25.1 

16.4 

34 

35 

2653.3 

2732.6 

2813.2 

2895.0 

2978.3 

3063.0 

3149.2 

3237.0 

3326.6 

3417.9 

35 

36 

54.7 

34.0 

14.5 

96.4 

79.7 

64.4 

50.6 

38.5 

28.1 

19.5 

36 

37 

56.0 

35.3 

15.9 

97.8 

81.1 

65.8 

52.1 

40.0 

29.6 

21.0 

37 

38 

57.3 

36.6 

17.2 

99.2 

82.5 

67.2 

53.5 

41.5 

31.1 

22.5 

38 

39 

58.6 

38.0 

18.6 

2900.5 

83.9 

68.7 

55.0 

42.9 

32.6 

24.1 

39 

40 

2659.9 

2739.3 

2820.0 

2901.9 

2985.3 

3070.1 

3156.4 

3244.4 

3334.1 

3425.6 

40 

41 

61.2 

40.6 

21.3 

03.3 

86.7 

71.5 

57.9 

45.9 

35.6 

27.2 

41 

42 

62.5 

42.0 

22.7 

04.7 

88.1 

72.9 

59.4 

47.4 

37.1 

28.7 

42 

43 

63.9 

43.3 

24.0 

06.1 

89.5 

74.4 

60.8 

48.9 

38.6 

30.2 

43 

44 

65.2 

44.6 

25.4 

07.4 

90.9 

75.8 

62.3 

50.3 

40.2 

31.8 

44 

45 

2666.5 

2746.0 

2826.7 

2908.8 

2992.3 

3077.2 

3163.7 

3251.8 

3341.7 

3433.3 

45 

46 

67.8 

47.3 

28.1 

10.2 

93.7 

78.7 

65.2 

53.3 

43.2 

34.9 

46 

47 

69.1 

48.6 

29.4 

11.6 

95.1 

80.1 

66.6 

54.8 

44.7 

36.4 

47 

48 

70.4 

50.0 

30.8 

13.0 

96.5 

81.5 

68.1 

56.3 

46.2 

38.0 

48 

49 

71.7 

51.3 

32.2 

14.3 

97.9 

82.9 

69.5 

57.8 

47.7 

39.5 

49 

50 

2673.1 

2752.7 

2833.5 

2915.7 

2999.3 

3084.4 

3171.0 

3259.3 

3349.2 

3441.0 

50 

51 

74.4 

54.0 

34.9 

17.1 

3000.7 

85.8 

72.5 

60.7 

50.8 

42.6 

51 

52 

75.7 

55.3 

36.2 

18.5 

02.1 

87.2 

73.9 

62.2 

52.3 

44.1 

52 

53 

77.0 

56.7 

37.6 

19.9 

03.5 

88.7 

75.4 

63.7 

53.8 

45.7 

53 

54 

78.3 

58.0 

39.0 

21.2 

04.9 

90.1 

76.8 

65.2 

55.3 

47.2 

54 

55 

2679.6 

2759.3 

2840.3 

2922.6 

3006.3 

3091.5 

3178.3 

3266.7 

3356.8 

3448.8 

55 

56 

81.0 

60.7 

41.7 

24.0 

07.7 

93.0 

79.7 

68.2 

58.3 

50.3 

56 

57 

82.3 

62.0 

43.0 

25.4 

09.2 

94.4 

81.2 

69.7 

59.9 

51.9 

57 

58 

83.6 

63.4 

44.4 

26.8 

10.6 

95.8 

82.7 

71.1 

61.4 

53.4 

58 

59 

84.9 

64.7 

45.8 

28.2 

12.0 

97.3 

84.1 

72.6 

62.9 

55.0 

59 

60 

2686.2 

2766.0 

2847.1 

2929.5 

3013.4 

3098.7 

3185.6 

3274.1 

3364.4 

3456.5 

60 

/ 

40° 

41° 

42° 

43° 

44° 

45° 

46° 

47° 

48° 

49° 

/ 

246 


Table  5.    Meridional  Parts 


1 

50° 

51° 

52° 

53° 

54° 

55° 

56° 

57° 

58° 

59° 

/ 

0 

3456.5 

3550.6 

3646.7 

3745.1 

3845.7 

3948.8 

4054.5 

4163.0 

4274.4 

4389.1 

0 

I 

58.1 

52.2 

48.4 

46.7 

47.4 

50.5 

56.3 

64.8 

76.3 

91.0 

1 

2 

59.6 

53.8 

50.0 

48.4 

49.1 

52.3 

58.1 

66.6 

78.2 

92.9 

2 

3 

61.2 

55.4 

51.6 

50.0 

50.8 

54.0 

59.8 

68.5 

80.1 

94.9 

3 

4 

62.7 

56.9 

53.2 

51.7 

52.5 

55.7 

61.6 

70.3 

82.0 

96.8 

4 

5 

3464.3 

3558.5 

3654.8 

3753.4 

3854.2 

3957.5 

4063.4 

4172.1 

4283.9 

4398.8 

5 

6 

65.9 

60.1 

56.5 

55.0 

55.9 

59.2 

65.2 

74.0 

85.7 

4400.7 

6 

7 

67.4 

61.7 

58:1 

56.7 

57.6 

61.0 

67.0 

75.8 

87.6 

02.6 

7 

8 

69.0 

63.3 

59.7 

58.3 

59.3 

62.7 

68.8 

77.7 

89.5 

04.6 

8 

9 

70.5 

64.9 

61.3 

60.0 

61.0 

64.5 

70.6 

79.5 

91.4 

06.5 

9 

10 

3472.1 

3566.5 

3663.0 

3761.7 

3862.7 

3966.2 

4072.4 

4181.3 

4293.3 

4408.5 

10 

11 

73.6 

68.1 

64.6 

63.3 

64.4 

68.0 

74.2 

83.2 

95.2 

10.4 

11 

12 

75.2 

69.7 

66.2 

65.0 

66.1 

69.7 

76.0 

85.0 

97.1 

12.4 

12 

13 

76.7 

71.3 

67.9 

66.7 

67.8 

71.5 

77.7 

86.9 

99.0 

14.3 

13 

14 

78.3 

72.8 

69.5 

68.3 

69.5 

73.2 

79.5 

88.7 

4300.9 

16.3 

14 

15 

3479.9 

3574.4 

3671.1 

3770.0 

3871.2 

3975.0 

4081.3 

4190.6 

4302.8 

4418.2 

15 

16 

81.4 

76.0 

72.7 

71.7 

72.9 

76.7 

83.1 

92.4 

04.7 

20.2 

16 

17 

83.0 

77.6 

74.4 

73.3 

74.6 

78.5 

84.9 

94.2 

06.6 

22.1 

17 

18 

84.5 

79.2 

76.0 

75.0 

76.3 

80.2 

86.7 

96.1 

08.5 

24.1 

18 

19 

86.1 

80.8 

77.6 

76.7 

78.1 

82.0 

88.5 

97.9 

10.4 

26.1 

19 

20 

3487.7 

3582.4 

3679.3 

3778.3 

3879.8 

3983.7 

4090.3 

4199.8 

4312.3 

4428.0 

20 

21 

89.2 

84.0 

80.9 

80.0 

81.5 

85.5 

92.1 

4201.6 

14.2 

30.0 

21 

22 

90.8 

85.6 

82.5 

81.7 

83.2 

87.2 

93.9 

03.5 

16.1 

31.9 

22 

23 

92.4 

87.2 

84.2 

83.3 

84.9 

89.0 

95.7 

05.3 

18.0 

33.9 

23 

24 

93.9 

88.8 

85.8 

85.0 

86.6 

90.7 

97.5 

07.2 

19.9 

35.8 

24 

25 

3495.5 

3590.4 

3687.4 

3786.7 

3888.3 

3992.5 

4099.3 

4209.0 

4321.8 

4437.8 

25 

26 

97.1 

92.0 

89.1 

88.4 

90.0 

94.3 

4101.1 

10.9 

23.7 

39.8 

26 

27 

98.6 

93.6 

90.7 

90.0 

91.8 

96.0 

02.9 

12.8 

25.6 

41.7 

27 

28 

3500.2 

95.2 

92.3 

91.7 

93.5 

97.8 

04.8 

14.6 

27.5 

43.7 

28 

29 

01.8 

96.8 

94.0 

93.4 

95.2 

99.5 

06.6 

16.5 

29.4 

45.7 

29 

30 

3503.3 

3598.4 

3695.6 

3795.1 

3896.9 

4001.3 

4108.4 

4218.3 

4331.3 

4447.0 

30 

31 

04.9 

3600.0 

97.3 

96.8 

98.6 

03.1 

10.2 

20.2 

33.2 

49.6 

31 

32 

06.5 

01.6 

98.9 

98.4 

3900.4 

04.8 

12.0 

22.0 

35.2 

51.6 

32 

33 

08.0 

03.2 

3700.5 

3800.1 

02.1 

06.6 

13.8 

23.9 

37.1 

53.5 

33 

34 

09.6 

04.8 

02.2 

01.8 

03.8 

08.3 

15.6 

25.8 

39.0 

55.5 

34 

35 

3511.2 

3606.4 

3703.8 

3803.5 

3905.5 

4010.1 

4117.4 

4227.6 

4340.9 

4457.5 

35 

36 

12.7 

08.0 

05.5 

05.1 

07.2 

11.9 

19.2 

29.5 

42.8 

59.4 

36 

37 

14.3 

09.6 

07.1 

06.8 

09.0 

13.6 

21.0 

31.3 

44.7 

61.4 

37 

38 

15.9 

11.2 

08.7 

08.5 

10.7 

15.4 

22.9 

33.2 

46.6 

63.4 

38 

39 

17.5 

12.8 

10.4 

10.2 

12.4 

17.2 

24.7 

35.1 

48.6 

65.4 

39 

40 

3519.0 

3614.5 

3712.0 

3811.9 

3914.1 

4018.9 

4126.5 

4236.9 

4350.5 

4467.3 

40 

41 

20.6 

16.1 

13.7 

13.6 

15.9 

20.7 

28.3 

38.8 

52.4 

69.3 

41 

42 

22.2 

17.7 

15.3 

15.2 

17.6 

22.5 

30.1 

40.7 

54.3 

71.3 

42 

43 

23.7 

19.3 

17.0 

17.0 

19.3 

24.3 

31.9 

42.5 

56.2 

73.3 

43 

44 

25.3 

20.9 

18.6 

18.6 

21.0 

26.0 

33.8 

44.4 

58.2 

75.3 

44 

45 

3526.9 

3622.5 

3720.3 

3820.3 

3922.8 

4027.8 

4135.6 

4246.3 

4360.1 

4477.2 

45 

46 

28.5 

24.1 

21.9 

22.0 

24.5 

29.6 

37.4 

48.1 

62.0 

79.2 

46 

47 

30.1 

25.7 

23.6 

23.7 

26.2 

31.4 

39.2 

50.0 

63.9 

81.2 

47 

48 

31.6 

27.3 

25.2 

25.4 

28.0 

33.1 

41.0 

51.9 

65.9 

83.2 

48 

49 

33.2 

29.0 

26.9 

27.1 

29.7 

34.9 

42.9 

53.8 

67.8 

85.2 

49 

50 

3534.8 

3630.6 

3728.5 

3828.7 

3931.4 

4036.7 

4144.7 

4255.6 

4369.7 

4487.2 

50 

51 

36.4 

32.2 

30.2 

30.4 

33.2 

38.5 

46.5 

57.5 

71.7 

89.1 

51 

52 

37.9 

33.8 

31.8 

32.1 

34.9 

40.2 

48.3 

59.4 

73.6 

91.1 

52 

53 

39.5 

35.4 

33.5 

33.8 

36.6 

42.0 

50.2 

61.3 

75.5 

93.1 

53 

54 

41.1 

37.0 

35.1 

35.5 

38.4 

43.8 

52.0 

63.1 

77.4 

95.1 

64 

55 

3542.7 

3038.6 

3736.8 

3837.2 

3940.1 

4045.6 

4153.8 

4265.0 

4379.4 

4497.1 

55 

56 

44.3 

40.3 

38.4 

38.9 

41.8 

47.4 

55.7 

66.9 

81.3 

99.1 

56 

57 

45.9 

41.9 

40.1 

40.6 

43.6 

49.1 

57.5 

68.8 

83.2 

4501.1 

57 

58 

47.4 

43.5 

41.7 

42.3 

45.3 

50.9 

59.3 

70.7 

85.2 

03.1 

58 

59 

49.C 

45.1 

43.4 

45.0 

47.0 

52.7 

61.1 

72.5 

87.1 

05.1 

59 

60 

3550.6 

3646.7 

3745.1 

3845.7 

3948.8 

4054.5 

4163.0 

4274.4 

4389.1 

4507.1 

60 

/ 

1  50° 

51° 

52° 

53° 

54° 

55° 

56° 

57° 

58° 

59° 

/ 

Table  6 


Table  7       247 


Combined  Correction  for  Observed 
Sextant  Altitudes 


Correction  for  Dip  of 

Sea  Horizon 

(Sun  or  Star) 


Observed 
Altitude 

Correction 

For   Sun    (to 
be  added   to 

observed  alti- 
tude) 

For  Star   (to 
be  subtracted 

from  observed 
altitude) 

5° 

6'  14" 

9'  55" 

6 

7   41 

8  28 

7 

8  45 

7  24 

8 

9  35 

6  34 

9 

10  16 

5  53 

10 

10  50 

5   19 

11 

11    17 

4  51 

12 

11  41 

4  27 

13 

12     2 

4     7 

14 

12   19 

3  49 

15 

12  34 

3  34 

20 

13  29 

2  39 

25 

14     3 

2     5 

30 

14  26 

1   41 

35 

14  44 

1   23 

40 

14  57 

1    10 

45 

15     8 

0  58 

50 

15   17 

0  49 

55 

15  25 

0  40 

60 

15  31 

0  34 

65 

15  37 

0  27 

70 

15  42 

0  21 

75 

15  47 

0   16 

80 

15  52 

0   10 

85 

15  55 

0     5 

Height  of 
Observer's 
Eye    Above 

Sea  Level 
(feet) 

Dip  Correc- 
tion (to  be 
subtracted 

from 
observed 
altitude) 

4 

1'  58" 

6 

2   24 

8 

2  46 

10 

3  06 

12 

3  24 

14 

3  40 

16 

3  55 

18 

4     9 

20 

4  23 

22 

4  36 

24 

4  48 

26 

5     0 

28 

5   11 

30 

5  22 

35 

5  48 

40 

6   12 

45 

6  36 

50 

6  56 

55 

7   16 

60 

7  35 

70 

8  12 

85 

9     2 

100 

9  48 

Small  supplementary  correction,  for  Sun 
only. 

Jan.  to  March  \  ,  ,  .pi,, 
andOct.  toDec./^ddlO  • 
April  to  Sept.,  subtract  10". 


The  dip  correction  is  not 
required  when  the  artificial 
horizon  is  used. 


248 


Table  8 


To  Change  Horn's  and  Minutes  into  Decimals  of  a  Day 


HOURS   EXPRESSED 

AS  DECIMAL  PARTS 

OF  A  DAY 


MINUTES  EXPRESSED  AS  DECIMAL  PARTS 
OF  A  DAY 


Hours 

Decimal 

1 

.0416 

2 

.0833 

3 

.1250 

4 

.1666 

5 

.2083 

6 

.2500 

7 

.2916 

8 

.3333 

9 

.3750 

10 

.4166 

11 

.4583 

12 

.5000 

13 

.5416 

14 

.5833 

15 

.6249 

16 

.6666 

17 

.7083 

18 

.7500 

19 

.7916 

20 

.8333 

21 

.8749 

22 

.9166 

23 

.9583 

24 

1.0000 

Minutes 

Decimal. 

Minutes 

Decimal 

1 

.0006 

31 

.0215 

2 

.0013 

32 

.0222 

3 

.0020 

33 

.0229 

4 

.0027 

34 

.0236 

5 

.0034 

35 

.0243 

6 

.0041 

36 

.0250 

7 

.0048 

37 

.0256 

8 

.0055 

38 

.0263 

9 

.0062 

39 

.0270 

10 

.0069 

40 

.0277 

11 

.0076 

41 

.0284 

12 

.0083 

42 

.0291 

13 

.0090 

43 

.0298 

14 

.0097 

44 

.0305 

15 

.0104 

45 

.0312 

16 

.0111 

46 

.0319 

17 

.0118 

47 

.0326 

18 

.0125 

48 

.0333 

19 

.0131 

49 

.0340 

20 

.0138 

50 

.0347 

21 

.0145 

51 

.0354 

22 

.0152 

52 

.0361 

23 

.0159 

53 

.0368 

24 

.0166 

54 

.0375 

25 

.0173 

55 

.0381 

26 

.0180 

56 

.0388 

27 

.0187 

57 

.0395 

28 

.0194 

58 

.0402 

29 

.0201 

59 

.0409 

30 

.0208 

60 

.0416 

Table  9 


249 


To  Interchange  Degrees  and  Minutes  of  Longitude  and  Hours,  Minutes^ 
and  Seconds  of  Time.    Part  1 


Oh 

I'' 

2h 

3^^ 

4h 

5^^ 

6^ 

7h 

Sh 

9h 

10^ 

11'' 

Qm 

0° 

15° 

30° 

45° 

60° 

75° 

90° 

105° 

120° 

135° 

150° 

165° 

4 

1 

16 

31 

46 

61 

76 

91 

106 

121 

136 

151 

166 

8 

2 

17 

32 

47 

62 

77 

92 

107 

122 

137 

152 

167 

12 

3 

18 

33 

48 

63 

78 

93 

108 

123 

138 

153 

168 

16 

4 

19 

34 

49 

64 

79 

94 

109 

124 

139 

154 

169 

20 

5 

20 

35 

50 

65 

80 

95 

110 

125 

140 

155 

170 

24 

6 

21 

36 

51 

66 

81 

96 

111 

126 

141 

156 

171 

28 

7 

22 

37 

52 

67 

82 

97 

112 

127 

142 

157 

172 

32 

8 

23 

38 

53 

68 

83 

98 

113 

128 

143 

158 

173 

36 

9 

24 

39 

54 

69 

84 

99 

114 

129 

144 

159 

174 

40 

10 

25 

40 

55 

70 

85 

100 

115 

130 

145 

160 

175 

44 

11 

26 

41 

56 

71 

86 

101 

116 

131 

146 

161 

176 

48 

12 

27 

42 

57 

72 

87 

102 

117 

132 

147 

162 

177 

52 

13 

28 

43 

58 

73 

88 

103 

118 

133 

148 

163 

178 

5^> 

14 

29 

44 

59 

74 

89 

104 

119 

134 

149 

164 

179 

12'' 

13^^ 

Mh 

15'' 

16'' 

I7h 

18'' 

19'' 

20'' 

21'^ 

22'' 

23'' 

Qm 

180° 

195° 

210° 

225° 

240° 

255° 

270° 

285° 

300° 

315° 

330° 

345° 

4 

181 

196 

211 

226 

241 

256 

271 

286 

301 

316 

331 

346 

8 

182 

197 

212 

227 

242 

257 

272 

287 

302 

317 

332 

347 

12 

183 

198 

213 

228 

243 

258 

273 

288 

303 

318 

333 

348 

16 

184 

199 

214 

229 

244 

259 

274 

289 

304 

319 

334 

349 

20 

185 

200 

215 

230 

245 

260 

275 

290 

305 

320 

335 

350 

24 

186 

201 

216 

231 

246 

261 

276 

291 

306 

321 

336 

351 

28 

187 

202 

217 

232 

247 

262 

277 

292 

307 

322 

337 

352 

32 

188 

203 

218 

233 

248 

263 

278 

293 

308 

323 

338 

353 

36 

189 

204 

219 

234 

249 

264 

279 

294 

309 

324 

339 

354 

40 

190 

205 

220 

235 

250 

265 

280 

295 

310 

325 

340 

355 

44 

191 

206 

221 

236 

251 

266 

281 

296 

311 

326 

341 

356 

48 

192 

207 

222 

237 

252 

267 

282 

297 

312 

327 

342 

357 

52 

193 

208 

223 

238 

253 

268 

283 

298 

313 

328 

343 

358 

56 

194 

209 

224 

239 

254 

269 

284 

299 

314 

329 

344 

359 

Part  2 


Explanation  of  Table  9 


3. 


1.  To  change  degrees  of  longitude  into  hours  and 
minutes  of  time:  Find  the  number  of  degrees  in  Part  1. 
The  required  hours  will  then  be  found  at  the  head  of  the 
column  containing  the  degrees,  and  the  required  min- 
utes at  the  left-hand  end  of  the  line  containing  the 
degrees. 

Examples:    113°  =  7'' 32'"  ;    294°  =  19'*  36^". 

2.  To  change  minutes  of  longitude  into  minutes  and 
seconds  of  time  :  Find  the  minutes  of  longitude  in  Part  2. 
The  required  minutes  and  seconds  of  time  will  again 
be  found  at  the  head  of  the  column  and  the  left-hand  end 
of  the  line. 

Examples :  43'  =  2"*  52« ;    28'  =  1"'  52'. 
1  and  2  can  be  combined  by  addition. 
Examples:    113°  43'  =  7h  34"'  52^ 
294°  28'  =  ig/i  37'"  52«. 
4.    To  change  hours  and  minutes  of  time  into  degrees 
and  minutes  of  longitude  :    Find  the  number  of  hours  at 
the  head  of  one  of  the  columns  of  Part  1  ;   then  run  down 
the  column  until  you  reach  a  line  having  at  its  left-hand 
end  a  number  of  minutes  equal  to  (or  just  smaller  than) 
the  given  number  of  minutes  of  time.     Where  that  line 
and  column  meet  you  will  find  the  required  degrees  of  longitude. 

Examples:    7'' 32'"  =  113°;    19't  36'"  =294°. 

5.  To  change  minutes  and  seconds  of  time  into  minutes  of  longitude  :  Find  the  number  of 
minutes  of  tirne  at  the  head  of  one  of  the  columns  of  Part  2  ;  then  run  down  the  column  until 
you  reach  a  line  having  at  its  left-hand  end  a  number  of  seconds  equal  (or  nearly  equal)  to 
the  given  number  of  seconds  of  time.  Where  that  line  and  column  meet  you  will  find  the 
minutes  of  longitude. 

Examples  :    2'"  52*  =  43' ;    1"'  52«  =  28'. 

6.  4  and  5  can  be  combined  by  addition : 

Examples:  7«34'"52'  =  113° 43';  19/'37'«52«  =294" 28'. 


Qm 

1'" 

2"' 

3m 

0^ 

0' 

15' 

30' 

45' 

4 

1 

16 

31 

46 

8 

2 

17 

32 

47 

12 

3 

18 

33 

48 

16 

4 

19 

34 

49 

20 

5 

20 

35 

50 

24 

6 

21 

36 

51 

28 

7 

22 

37 

52 

32 

8 

23 

38 

53 

36 

9 

24 

39 

54 

40 

10 

25 

40 

55 

44 

11 

26 

41 

56 

48 

12 

27 

42 

57 

52 

13 

28 

43 

58 

56 

14 

29 

44 

59 

250 


Table  10.    Hayersine  Table 


s    ' 

O''  0'"    0° 

Oh  J^r,.                lo 

Qh  8"^          2° 

Qh  12^          30   1 

Hav. 

No. 

Hav. 

No. 

Hav. 

No. 

Hav. 

No. 

0       0 

4  1 

5  2 
12       3 
16       4 
;^0   5 
^4   6 
28       7 
5:?   8 
36       9 
40  10 

44  11 
48    12 
5:^  13 
66     14 

2.32539 
.92745 

3.27963 
.52951 

3.72333 
.88169 

4.01559 
.13157 
.23388 

4.32539 
.40818 
.48375 
.55328 
.61765 

0.00000 
.00000 
.00000 
.00000 
.00000 

0.00000 
.00000 
.00000 
.00000 
.00000 

0.00000 
.00000 
.00000 
.00000 
.00000 

5.88168 
.89604 
.91016 
.92406 
.93774 

5.95121 
.96447 
.97753 
.99040 

6.00308 

6.01557 
.02789 
.04004 
.05202 
.06384 

0.00008 
.00008 
.00008 
.00008 
.00009 

0.00009 
.00009 
.00010 
.00010 
.00010 

0.00010 
.00011 
.00011 
.00011 
.00012 

6.48371 
.49092 
.49807 
.50516 
.51219 

6.51916 
.52608 
.53295 
.53976 
.54652 

6.55323 
.55988 
.56649 
.57304 
.57955 

0.00030 
.00031 
.00031 
.00032 
.00033 

0.00033 
.00034 
.00034 
.00035 
.00035 

0.00036 
.00036 
.00037 
.00037 
.00038 

6.83584 
.84065 
.84543 
.85019 
.85492 

6.85963 
.86431 
.86897 
.87360 
.87821 

6.88279 
.88735 
.89188 
.89639 
.90088 

0.00069 
.00069 
.00070 
.00071 
.00072 

0.00072 
.00073 
.00074 
.00075 
.00076 

0.00076 
.00077 
.00078 
.00079 
.00080 

s 

Qh  pn                QO 

Qh  gm                JO 

Qh  gm          2° 

Qh  13m          30 

0  15 

4  16 

5  17 
i^  18 
16     19 
^0  20 
24    21 
;g5  22 
5:^  23 
36     24 
40  25 

44  26 

45  27 
5;^  28 
56     29 

4.67757 
.73363 
.78629 
.83594 
.88290 

4.92745 
.96983 

5.01024 
.04885 
.08581 

5.12127 
.15534 
.18812 
.21971 
.25019 

0.00000 
.00001 
.00001 
.00001 
.00001 

0.00001 
.00001 
.00001 
.00001 
.00001 

0.00001 
.00001 
.00002 
.00002 
.00002 

6.07550 
.08700 
.09836 
.10956 
.12063 

6.13155 
.14234 
.15300 
.16353 
.17393 

6.18421 
.19437 
.20441 
.21433 
.22415 

0.00012 
.00012 
.00013 
.00013 
.00013 

0.00014 
.00014 
.00014 
.00015 
.00015 

0.00015 
.00016 
.00016 
.00016 
.00017 

6.58600 
.59241 
.59878 
.60509 
.61136 

6.61759 
.62377 
.62991 
.63600 
.64205 

6.64806 
.65403 
.65996 
.66585 
.67170 

0.00039 
.00039 
.00040 
.00040 
.00041 

0.00041 
.00042 
.00043 
.00043 
.00044 

0.00044 
.00045 
.00046 
.00046 
.00047 

6.90535 
.90979 
.91421 
.91860 
.92298 

6.92733 
.93166 
.93597 
.94026 
.94453 

6.94877 
.95300 
.95720 
.96139 
.96555 

0.00080 
.00081 
.00082 
.00083 
.00084 

0.00085 
.00085 
.00086 
.00087 
.00088 

0.00089 
.00090 
.00091 
.00091 
.00092 

s 

Qh  2'^           0° 

Qh  (jm               1° 

Qh  IQ'n             2° 

Qh  1/1,'n          30   1 

0  30 

4  31 

5  32 
12     33 
i5  34 
^0  35 
24     36 
:g5  37 
32     38 
55  39 
40  40 

44  41 

45  42 
62     43 
55  44 

5.27963 
.30811 
.33569 
.36242 
.38835 

5.41352 
.43799 
.46179 
.48496 
.50752 

5.52951 
.55095 
.57189 
.59232 
.61229 

0.00002 
.00002 
.00002 
.00002 
.00002 

0.00003 
.00003 
.00003 
.00003 
.00003 

0.00003 
.00004 
.00004 
.00004 
.00004 

6.23385 
.24345 
.25294 
.26233 
.27162 

6.28081 
.28991 
.29891 
.30781 
.31663 

6.32536 
.33400 
.34256 
.35103 
.35943 

0.00017 
.00018 
.00018 
.00018 
.00019 

0.00019 
.00019 
.00020 
.00020 
.00021 

0.00021 
.00022 
.00022 
.00022 
.00023 

6.67751 
.68328 
.68901 
.69470 
.70036 

6.70598 
.71157 
.71712 
.72263 
.72811 

6.73355 
.73896 
.74434 
.74969 
.75500 

0.00048 
.00048 
.00049 
.00050 
.00050 

0.00051 
.00051 
.00052 
.00053 
.00053 

0.00054 
.00055 
.00056 
.00056 
.00057 

6.96970 
.97382 
.97793 
.98201 
.98608 

6.99013 
.99416 
.99817 

7.00216 
.00613 

7.01009 
.01403 
.01795 
.02185 
.02573 

0.00093 
.00094 
.00095 
.00096 
.00097 

0.00098 
.00099 
.00100 
.00101 
.00101 

0.00102 
.00103 
.00104 
.00105 
.00106 

s 

Qh  gm                 QO 

Qh  7m     10 

Qhipn           2° 

Oh  16'"          3° 

0  45 

4  46 

5  47 
12     48 
i5  49 
20     50 
;^4  51 
;g5  52 
32     53 
35  54 
40     55 

44  56 

45  57 
62     58 
55  59 
60     60 

5.63181 
.65090 
.66958 
.68787 
.70578 

5.72332 
.74052 
.75739 
.77394 
.79017 

5.80611 
.82176 
.83713 
.85224 
.86709 

5.88168 

0.00004 
.00004 
.00005 
.00005 
.00005 

0.00005 
.00006 
.00006 
.00006 
.00006 

0.00006 
.00007 
.00007 
.00007 
.00007 

0.00008 

6.36774 
.37597 
.38412 
.39220 
.40021 

6.40814 
.41600 
.42379 
.43151 
.43916 

6.44675 
.45427 
.46172 
.46911 
.47644 

6.48371 

0.00023 
.00024 
.00024 
.00025 
.00025 

0.00026 
.00026 
.00027 
.00027 
.00027 

0.00028 
.00028 
.00029 
.00029 
.00030 

0.00030 

6.76028 
.76552 
.77074 
.77592 
.78108 

6.78620 
.79129 
.79630 
.80139 
.80640 

6.81137 
.81632 
.82124 
.82614 
.83100 

6.83584 

0.00058 
.00058 
.00059 
.00060 
.00060 

0.00061 
.00062 
.00063 
.00063 
.00064 

0.00065 
.00066 
.00066 
.00067 
.00068 

0.00069 

7.02960 
.03345 
.03729 
.04110 
.04490 

7.04869 
.05245 
.05620 
.05994 
.06366 

7.06736 
.07105 
.07472 
.07837 
.08201 

7.08564 

0.00107 
.00108 
.00109 
.00110 
.00111 

0.00112 
.00113 
.00114 
.00115 
.00116 

0.00117 
.00118 
.00119 
.00120 
.00121 

0.00122 

Table  10.    Haversine  Table 


251 


S     ' 

0^  i^'« 

4" 

Qh  20'"          5° 

Qh  24^          6° 

Qh  28"^          7° 

Hav. 

No. 

Hav. 

No. 

Hav. 

No. 

Hav. 

No. 

0       0 

4   1 
8       2 
12       3 
16       4 
20       5 
24       6 
28       7 
5^   8 
36       9 
40  10 

44  11 
48    12 
Jjg  13 
56     14 

7.08564 
.08925 
.09284 
.09642 
.09999 

7.10354 
.10708 
.11060 
.11411 
.11760 

7.12108 
.12455 
.12800 
.13144 
.13486 

0.00122 
.00123 
.00124 
.00125 
.00126 

0.00127 
.00128 
.00129 
.00130 
.00131 

0.00132 
.00133 
.00134 
.00135 
.00136 

7.27936 
.28225 
.28513 
.28800 
.29086 

7.29371 
.29655 
.29938 
.30220 
.30502 

7.30782 
.31062 
.31340 
.31618 
.31895 

0.00190 
.00192 
.00193 
.00194 
.00195 

0.00197 
.00198 
.00199 
.00201 
.00202 

0.00203 
.00204 
.00206 
.00207 
.00208 

7.43760 
.44001 
.44241 
.44480 
.44719 

7.44957 
.45194 
.45431 
.45667 
.45903 

7.46138 
.46372 
.46605 
.46838 
.47071 

0.00274 
.00275 
.00277 
.00278 
.00280 

0.00282 
.00283 
.00285 
.00286 
.00288 

0.00289 
.00291 
.00292 
.00294 
.00296 

7.57135 
.57341 
.57547 
.57752 
.57957 

7.58162 
.58366 
.58569 
.58772 
.58974 

7.59176 
.59378 
.59579 
.59779 
.59979 

0.00373 
.00374 
.00376 
.00378 
.00380 

0.00382 
.00383 
.00385 
.00387 
.00389 

0.00391 
.00392 
.00394 
.00396 
.00398 

s 

Qh  Ijm              4° 

Oh  21""          5° 

0^  25"^         6° 

Qh  29"^         7° 

0  15 

4  16 

5  17 
i;^  18 
16     19 
^0  20 
24     21 
^5  22 
S^  23 
36     24 
40  25 

44  26 

45  27 
5;g  28 
56     29 

7.13827 
.14167 
.14506 
.14843 
.15179 

7.15513 
.15846 
.16178 
.16509 
.16839 

7.17167 
.17494 
.17820 
.18144 
.18468 

0.00137 
.00139 
.00140 
.00141 
.00142 

0.00143 
.00144 
.00145 
.00146 
.00147 

0.00148 
.00150 
.00151 
.00152 
.00153 

7.32171 
.32440 
.32720 
.32994 
.33266 

7.33538 
.33809 
.34079 
.34348 
.34616 

7.34884 
.35150 
.35416 
.35681 
.35945 

0.00210 
.00211 
.00212 
.00214 
.00215 

0.00216 
.00218 
.00219 
.00221 
.00222 

0.00223 
.00225 
.00226 
.00227 
.00229 

7.47302 
.47533 
.47764 
.47994 
.48223 

7.48452 
.48680 
.48907 
.49134 
.49360 

7.49586 
.49811 
.50036 
.50259 
.50483 

0.00297 
.00299 
.00300 
.00302 
.00304 

0.00305 
.00307 
.00308 
.00310 
.00312 

0.00313 
.00315 
.00316 
.00318 
.00320 

7.60179 
.60378 
.60577 
.60775 
.60973 

7.61170 
.61367 
.61564 
.61760 
.61955 

7.62151 
.62345 
.62540 
.62733 
.62927 

0.00400 
.00402 
.00403 
.00405 
.00407 

0.00409 
.00411 
.00413 
.00415 
.00416 

0.00418 
.00420 
.00422 
.00424 
.00426 

s   ' 

Oh  J  8"'          4° 

Qh  22"'          5° 

Qh  26'"          6° 

Qh  SO"'          7° 

0     30 

4  31 

5  32 
12     33 
i6  34 
20     35 
^4  36 
^5  37 
32     38 
5^  39 
40     40 

44  41 

45  42 
52     43 
5^  44 

7.18790 
.19111 
.19430 
.19749 
.20066 

7.20383 
.20698 
.21012 
.21325 
.21636 

7.21947 
.22256 
.22565 
.22872 
.23178 

0.00154 
.00155 
.00156 
.00158 
.00159 

0.00160 
.00161 
.00162 
.00163 
.00165 

0.00166 
.00167 
.00168 
.00169 
.00171 

7.36209 
.36471 
.36733 
.36994 
.37254 

7.37514 
.37773 
.38030 
.38288 
.38544 

7.38800 
.39054 
.39309 
.39562 
.39815 

0.00230 
.00232 
.00233 
.00234 
.00236 

0.00237 
.00239 
.00240 
.00241 
.00243 

0.00244 
.00246 
.00247 
.00249 
.00250 

7.50706 
.50928 
.51149 
.51370 
.51591 

7.51811 
.52030 
.52249 
.52467 
.52685 

7.52902 
.53119 
.53335 
.53550 
.53766 

0.00321 
.00323 
.00325 
.00326 
.00328 

0.00330 
.00331 
.00333 
.00335 
.00336 

0.00338 
.00340 
.00341 
.00343 
.00345 

7.63120 
.63312 
.63504 
.63696 
.63887 

7.64078 
.64269 
.64458 
.64648 
.64837 

7.65026 
.65214 
.65402 
.65590 
.65777 

0.00428 
.00430 
.00432 
.00433 
.00435 

0.00437 
.00439 
.00441 
.00443 
.00445 

0.00447 
.00449 
.00451 
.00453 
.00455 

s 

Qh  lO'n             4° 

Qh  23"'          5° 

Qh  27"'          6° 

Qh  3im              70 

0     45 
4  46 
8     47 
i^  48 
16  49 
20     50 
;g4  51 
28     52 
f5^  53 
36  54 
40     55 
44  56 
48     57 
5:^  58 
56  59 
60     60 

7.23483 
.23787 
.24090 
.24392 
.24693 

7.24993 
.25292 
.25590 
.25886 
.26182 

7.26477 
.26771 
.27064 
.27355 
.27646 

7.27936 

0.00172 
.00173 
.00174 
.00175 
.00177 

0.00178 
.00179 
.00180 
.00181 
.00183 

0.00184 
.00185 
.00186 
.00188 
.00189 

0.00190 

7.40067 
.40318 
.40568 
.40818 
.41067 

7.41315 
.41563 
.41810 
.42056 
.42301 

7.42546 
.42790 
.43034 
.43277 
.43519 

7.43760 

0.00252 
.00253 
.00255 
.00256 
.00257 

0.00259 
.00260 
.00262 
.00263 
.00265 

0.00266 
.00268 
.00269 
.00271 
.00272 

0.00274 

7.53980 
.54194 
.54407 
.54620 
.54833 

7.55045 
.55256 
.55467 
.55677 
.55887 

7.56096 
.56305 
.56513 
.56721 
.56928 

7.57135 

0.00347 
.00348 
.00350 
.00352 
.00353 

0.00355 
.00357 
.00359 
.00360 
.00362 

0.00364 
.00366 
.00367 
.00369 
.00371 

0.00373 

7.65964 
.66150 
.66336 
.66521 
.66706 

7.66891 
.67075 
.67259 
.67443 
.67626 

7.67809 
.67991 
.68173 
.68355 
.68536 

7.68717 

0.00457 
.00459 
.00461 
.00463 
.00465 

0.00467 
.00469 
.00471 
.00473 
.00475 

0.00477 
.00479 
.00481 
.00483 
.00485 

0.00487 

252 


Table  10.    Haversine  Table 


s 

f)A  5^m 

8° 

Oh  36'" 

9° 

Oh  40"' 

10° 

Oh  44^ 

11° 

Hav. 

No. 

Hav. 

7.78929 

No. 

Hav. 

No. 
0.00760 

Hav. 

No. 

0       0 

7.68717 

0.00487 

0.00616 

7.88059 

7.96315 

0.00919 

4      1 

.68897 

.00489 

.79089 

.00618 

.88203 

.00762 

.96446 

.00921 

8       2 

.69077 

.00491 

.79249 

.00620 

.88348 

.00765 

.96577 

.00924 

12       3 

.69257 

.00493 

.79409 

.00622 

.88491 

.00767 

.96707 

.00927 

16       4 

.69437 

.00495 

.79568 

.00625 

.88635 

.00770 

.96838 

.00930 

20       5 

7.69616 

0.00497 

7.79728 

0.00627 

7.88778 

0.00772 

7.96968 

0.00933 

24       6 

.69794 

.00499 

.79886 

.00629 

.88921 

.00775 

.97098 

.00935 

28       7 

.69972 

.00501 

.80045 

.00632 

.89064 

.00777 

.97228 

.00938 

5^   8 

.70150 

.00503 

.80203 

.00634 

.89207 

.00780 

.97358 

.00941 

5(?   9 

.70328 

.00505 

.80361 

.00636 

.89349 

.00783 

.97478 

.00944 

40    10 

7.70505 

0.00507 

7.80519 

0.00639 

7.89491 

0.00785 

7.97617 

0.00947 

44  11 

.70682 

.00509 

.80677 

.00641 

.89633 

.00788 

.97746 

.00949 

48    12 

.70858 

.00511 

.80834 

.00643 

.89775 

.00790 

.97875 

.00952 

Jji?  13 

.71034 

.00513 

.80991 

.00646 

.89916 

.00793 

.98003 

.00955 

oG  14 

.71210 

.00515 

.81147 

.00648 

.90057 

.00795 

.98132 

.00958 

s   ' 

Oh  g^m 

8° 

Oh  37m 

9° 

Oh  J^pn 

10° 

Oh  45"' 

11° 

0     15 

7.71385 

0.00517 

7.81303 

0.00650 

7.90198 

0.00798 

7.98260 

0.00961 

4  16 

.71560 

.00520 

.81459 

.00653 

.90339 

.00801 

.98389 

.00964 

/?  17 

.71735 

.00522 

.81615 

.00655 

.90480 

.00803 

.98517 

.00966 

;;^  18 

.71909 

.00524 

.81771 

.00657 

.90620 

.00806 

.98644 

.00969 

16     19 

.72083 

.00526 

.81926 

.00660 

.90760 

.00808 

.98772 

.00972 

;gO  20 

7.72257 

0.00528 

7.82081 

0.00662 

7.90900 

0.00811 

7.98899 

0.00975 

24     21 

.72430 

.00530 

.82235 

.00664 

.91039 

.00814 

.99027 

.00978 

^5  22 

.72603 

.00532 

.82390 

.00667 

.91179 

.00816 

.99154 

.00981 

5^  23 

.72775 

.00534 

.82544 

.00669 

.91318 

.00819 

.99281 

.00984 

36    24 

.72948 

.00536 

.82698 

.00671 

.91457 

.00821 

.99407 

.00986 

40  25 

7.73119 

0.00539 

7.82851 

0.00674 

7.91596 

0.00824 

7.99534 

0.00989 

44    26 

.73291 

.00541 

.83004 

.00676 

.91734 

.00827 

.99660 

.00992 

45  27 

.73462 

.00543 

.83157 

.00679 

.91872 

.00829 

.99786 

.00995 

5^  28 

.73633 

.00545 

.83310 

.00681 

.92010 

.00832 

.99912 

.00998 

36     29 

.73803 

.00547 

.83463 

.00683 

.92148 

.00835 

8.00038 

.01001 

s 

Oh  34"" 

8° 

Qh  38"" 

9° 

Oh  42"' 

10° 

(^46"' 

11° 

0  30 

7.73974 

0.00549 

7.83615 

0.00686 

7.92286 

0.00837 

8.00163 

0.01004 

4  31 

.74143 

.00551 

.83767 

.00688 

.92423 

.00840 

.00289 

.01007 

<§  32 

.74313 

.00554 

.83918 

.00691 

.92560 

.00843 

.00414 

.01010 

12     33 

.74482 

.00556 

.84070 

.00693 

.92697 

.00845 

.00539 

.01012 

i^  34 

.74651 

00558 

.84221 

.00695 

.92834 

.00848 

.00664 

.01015 

:gO  35 

7.74819 

0.00560 

7.84372 

0.00698 

7.92970 

0.00851 

8.00788 

0.01018 

24     36 

.74988 

.00562 

.84522 

.00700 

.93107 

.00853 

.00913 

.01021 

^5  37 

.75155 

.00564 

.84672 

.00703 

.93243 

.00856 

.01037 

.01024 

32     38 

.75323 

.00567 

.84822 

.00705 

.93379 

.00859 

.01161 

.01027 

55  39 

.75490 

.00569 

.84972 

.00707 

.93514 

.00861 

.01285 

.01030 

40  40 

7.75657 

0.00571 

7.85122 

0.00710 

7.93650 

0.00864 

8.01409 

0.01033 

44    41 

.75824 

.00573 

.85271 

.00712 

.93785 

.00867 

.01532 

.01036 

45  42 

.75990 

.00575 

.85420 

.00715 

.93920 

.00869 

.01656 

.01039 

52     43 

.76156 

.00578 

.85569 

.00717 

.94055 

.00872 

.01779 

.01042 

55  44 

.76321 

.00580 

.85717 

.00720 

.94189 

.00875 

.01902 

.01045 

s 

OA  35"' 

8° 

Oh  sgm 

9° 

Oh  43"' 

10° 

Oh  47"' 

11° 

0     45 

7.76487 

0.00582 

7.85866 

0.00722 

7.94324 

0.00877 

8.02025 

0.01048 

4  46 

.76652 

.00584 

.86014 

.00725 

.94458 

.00880 

.02148 

.01051 

5  47 

.76816 

.00586 

.86161 

.00727 

.94592 

.00883 

.02270 

.01054 

12     48 

.76981 

.00589 

.86309 

.00730 

.94726 

.00886 

.02392 

.01057 

15  49 

.77145 

.00591 

.86456 

.00732 

.94859 

.00888 

.02515 

.01060 

20     50 

7.77308 

0.00593 

7.86603 

0.00735 

7.94992 

0.00891 

8.02637 

0.01063 

^4  51 

.77472 

.00595 

.86750 

.00737 

.95126 

.00894 

.02758 

.01066 

^5  52 

.77635 

.00598 

.86896 

.00740 

.95259 

.00897 

.02880 

.01069 

32     53 

.77798 

.00600 

.87042 

.00742 

.95391 

.00899 

.03001 

.01072 

55  54 

.77960 

.00602 

.87188 

.00745 

.95524 

.00902 

.03123 

.01075 

40     55 

7.78122 

0.00604 

7.87334 

0.00747 

7.95656 

0.00905 

8.03244 

0.01078 

44  56 

.78284 

.00607 

.87480 

.00750 

.95788 

.00908 

.03365 

.01081 

45  57 

.78446 

.00609 

.87625 

.00752 

.95920 

.00910 

.03486 

.01084 

52     58 

.78607 

.00611 

.87770 

.00755 

.96052 

.00913 

.03606 

.01087 

55  59 

.78768 

.00613 

.87915 

.00757 

.96183 

.00916 

.03727 

.01090 

60     60 

7.78929 

0.00616 

7.88059 

0.00760 

7.96315 

0.00919 

8.03847 

0.01093 

Table  10.    Haversine  Table 


253 


s 

Qh  48"^ 

12° 

Qh  s2"' 

13° 

Oh  56'"- 

14° 

IhQm 

15° 

Hav. 

No. 

Hav. 

8.10772 

No. 

Hav. 

8.17179 

No. 

Hav. 

No. 

0       0 

8.03847 

0.01093 

0.01282 

0.01485 

8.23140 

0.01704 

4      1 

.03967 

.01096 

.10883 

.01285 

.17282 

.01489 

.23235 

.01707 

8       2 

.04087 

.01099 

.10993 

.01288 

.17384 

.01492 

.23331 

.01711 

12       3 

.04207 

.01102 

.11104 

.01291 

.17487 

.01496 

.23427 

.01715 

16       4 

.04326 

.01105 

.11214 

.01295 

.17590 

.01499 

.23523 

.01719 

20       5 

8.04446 

0.01108 

8.11324 

0.01298 

8.17692 

0.01503 

8.23618 

0.01723 

24       6 

.04565 

.01111 

.11435 

.01301 

.17794 

.01506 

.23713 

.01726 

28       7 

.04684 

.01114 

.11544 

.01305 

.17896 

.01510 

.23809 

.01730 

5^   8 

.04803 

.01117 

.11654 

.01308 

.17998 

.01513 

.23904 

.01734 

55   9 

.04922 

.01120 

.11764 

.01311 

.18100 

.01517 

.23999 

.01738 

40    10 

8.05041 

0.01123 

8.11873 

0.01314 

8.18202 

0.01521 

8.24094 

0.01742 

44  11 

.05159 

.01126 

.11983 

.01317 

.18303 

.01524 

.24189 

.01745 

48     12 

.05277 

.01129 

.12092 

.01321 

.18405 

.01528 

.24283 

.01749 

5^  13 

.05395 

.01132 

.12201 

.01324 

.18506 

.01531 

.24378 

.01753 

55  14 

.05513 

.01135 

.12310 

.01328 

.18607 

.01535 

.24473 

Jh  pn 

.01757 
15° 

s 

Oh  A9"' 

12° 

Qh  o3"' 

13° 

Oh  57'" 

14° 

0     15 

8.05631 

0.01138 

8.12419 

0.01331 

8.18709 

0.01538 

8.24567 

0.01761 

4  16 

.05749 

.01142 

.12528 

.01334 

.18810 

.01542 

.24661 

.01764 

8     17 

.05866 

.01145 

.12636 

.01338 

.18910 

.01546 

.24755 

.01768 

i^  18 

.05984 

.01148 

.12745 

.01341 

.19011 

.01549 

.24850 

.01772 

16     19 

.06101 

.01151 

.12853 

.01344 

.19112 

.01553 

.24944 

.01776 

:gO  20 

8.06218 

0.01154 

8.12961 

0.01348 

8.19212 

0.01556 

8.25037 

0.01780 

^4  21 

.06335 

.01157 

.13069 

.01351 

.19313 

.01560 

.25131 

.01784 

^5  22 

.06451 

.01160 

.13177 

.01354 

.19413 

.01564 

.25225 

.01788 

5^  23 

.06568 

.01163 

.13285 

.01358 

.19513 

.01567 

.25319 

.01791 

36     24 

.06684 

.01166 

.13392 

.01361 

.19613 

.01571 

.25412 

.01795 

40  25 

8.06800 

0.01170 

8.13500 

0.01365 

8.19713 

0.01574 

8.25505 

0.01799 

44  26 

.06917 

.01173 

.13607 

.01368 

.19813 

.01578 

.25599 

.01803 

48     27 

.07032 

.01176 

.13714 

.01371 

.19913 

.01582 

.25692 

.01807 

5;e  28 

.07148 

.01179 

.13822 

.01375 

.20012 

.01585 

.25785 

.01811 

56     29 

.07264 

.01182 

.13928 

.01378 

.20112 

.01589 

.25878 

Ih  2^ 

.01815 
15° 

s 

Oh  50"" 

12° 

Oh  54"" 

13° 

Oh  58'"- 

14° 

0  30 

8.07379 

0.01185 

8.14035 

0.01382 

8.20211 

0.01593 

8.25971 

0.01818 

4  31 

.07494 

.01188 

.14142 

.01385 

.20310 

.01596 

.26064 

.01822 

5  32 

.07610 

.01192 

.14248 

.01388 

.20410 

.01600 

.26156 

.01826 

;^  33 

.07725 

.01195 

.14355 

.01392 

.20509 

.01604 

.26249 

.01830 

16     34 

.07839 

.01198 

.14461 

.01395 

.20608 

.01607 

.26341 

.01834 

^0  35 

8.07954 

0.01201 

8.14567 

0.01399 

8.20706 

0.01611 

8.26434 

0.01838 

24     36 

.08069 

.01204 

.14673 

.01402 

.20805 

.01615 

.26526 

.01842 

^5  37 

.08183 

.01207 

.14779 

.01405 

.20904 

.01618 

.26618 

.01846 

3^  38 

.08297 

.01211 

.14885 

.01409 

.21002 

.01622 

.26710 

.01850 

36     39 

.08411 

.01214 

.14991 

.01412 

.21100 

.01626 

.26802 

.01854 

40  40 

8.08525 

0.01217 

8.15096 

0.01416 

8.21199 

0.01629 

8.26894 

0.01858 

44     41 

.08639 

.01220 

.15201 

.01419 

.21297 

.01633 

.26986 

.01861 

4<5  42 

.08752 

.01223 

.15307 

.01423 

.21395 

.01637 

.27078 

.01865 

5^  43 

.08866 

.01226 

.15412 

.01426 

.21493 

.01640 

.27169 

.01869 

56     44 

.08979 

.01230 

.15517 

.01429 

.21590 

.01644 

.27261 

.01873 

s 

Qh  gjm 

12° 

Oh  55"" 

13° 

Qh  grjm 

14° 

Jh  gm 

8.27352 

15° 
0.01877 

0  45 

8.09092 

0.01233 

8.15622 

0.01433 

8.21688 

0.01648 

4  46 

.09205 

.01236 

.15726 

.01436 

.21785 

.01651 

.27443 

.01881 

5  47 

.09318 

.01239 

.15831 

.01440 

.21883 

.01655 

.27534 

.01885 

12     48 

.09431 

.01243 

.15935 

.01443 

.21980 

.01659 

.27626 

.01889 

i5  49 

.09543 

.01246 

.16040 

.01447 

.22077 

.01663 

.27717 

.01893 

^0  50 

8.09656 

0.01249 

8.16144 

0.01450 

8.22175 

0.01666 

8.27807 

0.01897 

24     51 

.09768 

.01252 

.16248 

.01454 

.22272 

.01670 

.27898 

.01901 

^5  52 

.09880 

.01255 

.16352 

.01457 

.22368 

.01674 

.27989 

.01905 

32     53 

.09992 

.01259 

.16456 

.01461 

.22465 

.01677 

.28080 

.01909 

55  54 

.10104 

.01262 

.16559 

.01464 

.22562 

.01681 

.28170 

.01913 

40  55 

8.10216 

0.01265 

8.16663 

0.01468 

8.22658 

0.01685 

8.28260 

0.01917 

44     56 

.10327 

.01268 

.16766 

.01471 

.22755 

.01689 

.28351 

.01921 

45  57 

.10439 

.01272 

.16870 

.01475 

.22851 

.01692 

.28441 

.01925 

52     58 

.10550 

.01275 

.16973 

.01478 

.22947 

.01696 

.28531 

.01929 

.56'  59 

.10661 

.01278 

.17076 

.01482 

.23044 

.01700 

.28621 

.01933 

50  60 

8.10772 

0.01282 

8.17179 

0.01485 

8.23140 

0.01704 

8.28711 

0.01937 

254 


Table  10.    Haversine  Table 


s 

Ih  jm 

16° 

Ih  8'" 

17° 

Ih  12"^ 

18° 

Ih  16'n 

19° 

Hav. 

No. 

Hav. 

No. 

Hav. 

8.38867 

No. 
0.02447 

Hav. 

No. 

0       0 

8.28711 

0.01937 

8.33940 

0.02185 

8.43522 

0.02724 

4   1 

.28801 

.01941 

.34025 

.02189 

.38946 

.02452 

.43597 

.02729 

8       2 

.28891 

.01945 

.34109 

.02193 

.39026 

.02456 

.43673 

.02734 

12       3 

.28980 

.01949 

.34194 

.02198 

.39105 

.02461 

.43748 

.02738 

16       4 

.29070 

.01953 

.34278 

.02202 

.39185 

.02465 

.43823 

.02743 

20       5 

8.29159 

0.01957 

8.34362 

0.02206 

8.39264 

0.02470 

8.43899 

0.02748 

24       6 

.29249 

.01961 

.34446 

.02210 

.39344 

.02474 

.43974 

.02753 

28       7 

.29338 

.01965 

.34530 

.02215 

.39423 

.02479 

.44049 

.02757 

S^   8 

.29427 

.01969 

.34614 

.02219 

.39502 

.02483 

.44124 

.02762 

35   9 

.29516 

.01973 

.34698 

.02223 

.39581 

.02488 

.44199 

.02767 

40    10 

8.29605 

0.01977 

8.34782 

0.02227 

8.39660 

0.02492 

8.44273 

0.02772 

44  11 

.29694 

.01981 

.34865 

.02232 

.39739 

.02497 

.44348 

.02776 

48     12 

.29783 

.01985 

.34949 

.02236 

.39818 

.02501 

.44423 

.02781 

5^  13 

.29872 

.01989 

.35032 

.02240 

.39897 

.02506 

.44498 

.02786 

5^  14 

.29960 

.01993 

.35116 

.02245 

.39976 

.02510 

.44572 

.02791 

s 

Ih  s'" 

16° 

Ih  gm 

17° 

Ih  IS'"" 

18° 

Ih  17"" 

19° 

0     15 

8.30049 

0.01998 

8.35199 

0.02249 

8.40055 

0.02515 

8.44647 

0.02796 

4  16 

.30137 

.02002 

.35282 

.02253 

.40133 

.02520 

.44721 

.02800 

8     17 

.30226 

.02006 

.35365 

.02258 

.40212 

.02524 

.44796 

.02805 

i^  18 

.30314 

.02010 

.35449 

.02262 

.40290 

.02529 

.44870 

.02810 

16     19 

.30402 

.02014 

.35532 

.02266 

.40369 

.02533 

.44944 

.02815 

;gO  20 

8.30490 

0.02018 

8.35614 

0.02271 

8.40447 

0.02538 

8.45018 

0.02820 

;g4  21 

.30578 

.02022 

.35697 

.02275 

.40525 

.02542 

.45093 

.02824 

28     22 

.30666 

.02026 

.35780 

.02279 

.40603 

.02547 

.45167 

.02829 

5^  23 

.30754 

.02030 

.35863 

.02284 

.40681 

.02552 

.45241 

.02834 

36     24 

.30842 

.02034 

.35945 

.02288 

.40760 

.02556 

.45315 

.02839 

40  25 

8.30929 

0.02038 

8.36028 

0.02292 

8.40837 

0.02561 

8.45388 

0.02844 

44  26 

.31017 

.02043 

.36110 

.02297 

.40915 

.02565 

.45462 

.02849 

48     27 

.31104 

.02047 

.36193 

.02301 

.40993 

.02570 

.45536 

.02853 

5^  28 

.31192 

.02051 

.36275 

.02305 

.41071 

.02575 

.45610 

.02858 

56     29 

.31279 

.02055 

.36357 

.02310 

.41149 

.02579 

.45683 

.02863 

s 

Jh  6>n 

16° 

Ih  10'"- 

17° 

Ih  14'^ 

18° 

Ih  IS"' 

19° 

0  30 

8.31366 

0.02059 

8.36439 

0.02314 

8.41226 

0.02584 

8.45757 

0.02868 

4  31 

.31453 

.02063 

.36521 

.02319 

.41304 

.02588 

.45830 

.02873 

5  32 

.31540 

.02067 

.36603 

.02323 

.41381 

.02593 

.45904 

.02878 

/^  33 

.31627 

.02071 

.36685 

.02327 

.41459 

.02598 

.45977 

.02883 

16     34 

.31714 

.02076 

.36767 

.02332 

.41536 

.02602 

.46050 

.02887 

^0  35 

8.31800 

0.02080 

8.36849 

0.02336 

8.41613 

0.02607 

8.46124 

0.02892 

24     36 

.31887 

.02084 

.36930 

.02340 

.41690 

.02612 

.46197 

.02897 

:g5  37 

.31974 

.02088 

.37012 

.02345 

.41767 

.02616 

.46270 

.02902 

5^  38 

.32060 

.02092 

.37093 

.02349 

.41845 

.02621 

.46343 

.02907 

36     39 

.32147 

.02096 

.37175 

.02354 

.41921 

.02626 

.46416 

.02912 

4f?  40 

8.32233 

0.02101 

8.37256 

0.02358 

8.41998 

0.02630 

8.46489 

0.02917 

44    41 

.32319 

.02105 

.37337 

.02363 

.42075 

.02635 

.46562 

.  02922 

45  42 

.32405 

.02109 

.37419 

.02367 

.42152 

.02639 

.46634 

.02926 

5^  43 

.32491 

.02113 

.37500 

.02371 

.42229 

.02644 

.46707 

.02931 

56     44 

.32577 

.02117 

.37581 

.02376 

.42305 

.02649 

.46780 

.02936 

s 

2h  jrn 

16° 

Ih  11'^ 

17° 

Ih  13"" 

18° 

Ih  19'^ 

19° 

0  45 

8.32663 

0.02121 

8.37662 

0.02380 

8.42382 

0.02653 

8.46852 

0.02941 

4  46 

.32749 

.02126 

.37742 

.02385 

.42458 

.02658 

.46925 

.02946 

5  47 

.32834 

.02130 

.37823 

.02389 

.42535 

.02663 

.46998 

.02951 

12     48 

.32920 

.02134 

.37904 

.02394 

.42611 

.02668 

.47070 

.02956 

ig  49 

.33006 

.02138 

.37985 

.02398 

.42687 

.02672 

.47142 

.02961 

^0  50 

8.33091 

0.02142 

8.38065 

0.02402 

8.42764 

0.02677 

8.47215 

0.02966 

24     51 

.33176 

.02147 

.38146 

.02407 

.42840 

.02682 

.47287 

.02971 

^5  52 

.33262 

.02151 

.38226 

.02411 

.42916 

.02686 

.47359 

.02976 

32     53 

.33347 

.02155 

.38306 

.02416 

.42992 

.02691 

.47431 

.02981 

55  54 

.33432 

.02159 

.38387 

.02420 

.43068 

.02696 

.47503 

.02986 

40  55 

8.33517 

0.02164 

8.38467 

0.02425 

8.43144 

0.02700 

8.47575 

0.02991 

44     56 

.33602 

.02168 

.38547 

.02429 

.43219 

.02705 

.47647 

.02996 

45  57 

.33686 

.02172 

.38627 

.02434 

.43295 

.02710 

.47719 

.03000 

52     58 

.33771 

.02176 

.38707 

.02438 

.43371 

.02715 

.47791 

.03005 

55  59 

.33856 

.02181 

.38787 

.02443 

.43446 

.02719 

.47862 

.03010 

50  60 

8.33940 

0.02185 

8.38867 

0.02447 

8.43522 

0.02724 

8.47934 

0.03015 

Table  10.    Haversine  Table 


255 


s 

P^  21J"' 

20° 

Ih  24'" 

21° 

Ih  28'" 

22° 

Jh  S2'" 

23° 

Hav. 

No. 

Hav. 

No. 

Hav. 

No. 

Hav. 

8.59931 

No. 

0       0 

8.47934 

0.03015 

8.52127 

0.03321 

8.56120 

0.03641 

0.03975 

4   1 

.48006 

.03020 

.52195 

.03326 

.56185 

.03646 

.59993 

.03980 

8       2 

.48077 

.03025 

.52263 

.03331 

.56250 

.03652 

.60055 

.03986 

12       3 

.48149 

.03030 

.52331 

.03337 

.56315 

.03657 

.60117 

.03992 

16       4 

.48220 

.03035 

.52399 

.03342 

.56379 

.03663 

.60179 

.03998 

20       5 

8.48292 

0.03040 

8.52467 

0.03347 

8.56444 

0.03668 

8.60241 

0.04003 

24       6 

.4836.3 

.03045 

.52535 

.03352 

.56509 

.03674 

.60303 

.04009 

28       7 

.48434 

.03050 

.52602 

.03358 

.56574 

.03679 

.60365 

.04015 

.'J^   8 

.48505 

.03055 

.52670 

.03363 

.56638 

.03685 

.60426 

.04020 

5^   9 

.48576 

.03060 

.52738 

.03368 

.56703 

.03690 

.60488 

.04026 

40     10 

8.48648 

0.03065 

8.52806 

0.03373 

8.56767 

0.03695 

8.60550 

0.04032 

44  11 

.48719 

.03070 

.52873 

.03379 

.56832 

.03701 

.60611 

.04038 

48     12 

.48789 

.03075 

.52941 

.03384 

.56896 

.03706 

.60673 

.04043 

J^  13 

.48860 

.03080 

.53008 

.03389 

.56960 

.03712 

.60734 

.04049 

5^  14 

.48931 

.03085 

.53076 

.03394 

.57025 

.03717 

.60796 

V^  33'" 

.04055 
23° 

s 

Ih  21'" 

20° 

Ih   o2,^m 

21° 

Ih  29'" 

22° 

0     15 

8.49002 

0.03090 

8.53143 

0.03400 

8.57089 

0.03723 

8.60857 

0.04060 

4  16 

.49073 

.03095 

.53210 

.03405 

.57153 

.03728 

.60919 

.04066 

8     17 

.49143 

.03101 

.53277 

.03410 

.57217 

.03734 

.60980 

.04072 

i^  18 

.49214 

.03106 

.53345 

.03415 

.57282 

.03740 

.61041 

.04078 

16     19 

.49284 

.03111 

.53412 

.03421 

.57346 

.03745 

.61103 

.04083 

^0  20 

8.49355 

0.03116 

8.53479 

0.03426 

8.57410 

0.03751 

8.61164 

0.04089 

^4  21 

.49425 

.03121 

.53546 

.03431 

.57474 

.03756 

.61225 

.04095 

28     22 

.49490 

.03126 

.53613 

.03437 

.57538 

.03762 

.61286 

.04101 

5^  23 

.49566 

.03131 

.53680 

.03442 

.57601 

.03767 

.61347 

.04106 

36     24 

.49636 

.03136 

.53747 

.03447 

.57665 

.03773 

.61408 

.04112 

40  25 

8.49706 

0.03141 

8.53814 

0.03453 

8.57729 

0.03778 

8.61469 

0.04118 

44  26 

.49777 

.03146 

.53880 

.03458 

.57793 

.03784 

.61530 

.04124 

48     27 

.49847 

.03151 

.53947 

.03463 

.57856 

.03789 

.61591 

.04130 

5;^  28 

.49917 

.03156 

.54014 

.03468 

.57920 

.03795 

.61652 

.04135 

56     29 

s 

.49987 

.03161 

.54080 

.03474 

.57984 

.03800 

.61713 

Ih  34'" 

.04141 
23° 

jh  22'" 

20° 

Ih  26"' 

21° 

Ih  SO'" 

22° 

0  30 

8.50056 

0.03166 

8.54147 

0.03479 

8.58047 

0.03806 

8.61773 

0.04147 

4    31 

.50126 

.03171 

.54214 

.03484 

.58111 

.03812 

.61834 

.04153 

^  32 

.50196 

.03177 

.54280 

.03490 

.58174 

.03817 

.61895 

.04159 

i^  33 

.50266 

.03182 

.54346 

.03495 

.58238 

.03823 

.61955 

.04164 

16     34 

.50335 

.03187 

.54413 

.03500 

.58301 

.03828 

.62016 

.04170 

^0  35 

8.50405 

0.03192 

8.54479 

0.03506 

8.58364 

0.03834 

8.62077 

0.04176 

24     36 

.50475 

.03197 

.54545 

.03511 

.58427 

.03839 

.62137 

.04182 

^5  37 

.50544 

.03202 

.54612 

.03517 

.58491 

.03845 

.62197 

.04188 

5^  38 

.50614 

.03207 

.54678 

.03522 

.58554 

.03851 

.62258 

.04194 

36     39 

.50683 

.03212 

.54744 

.03527 

.58617 

.03856 

.62318 

.04199 

40  40 

8.50752 

0.03218 

8.54810 

0.03533 

8.58680' 

0.03862 

8.62379 

0.04205 

44    41 

.50821 

.03223 

.54876 

.03538 

.58743 

.03867 

.62439 

.04211 

4<§  42 

.50891 

.03228 

.54942 

.03543 

.58806 

.03873 

.62499 

.04217 

5^  43 

.50960 

.03233 

.55008 

.03549 

.58869 

.03879 

.62559 

.04223 

56     44 

.51029 

.03238 

.55073 

.03554 

.58932 

.03884 

.62619 

.04229 

s 

Jh  23'" 

20° 

Ih  27'" 

21° 

Jh  3Jm 

22° 

Ih  35'" 

23° 

0  45 

8.51098 

0.03243 

8.55139 

0.03560 

8.58994 

0.03890 

8.62680 

0.04234 

4     46 

.51167 

.03248 

.55205 

.03565 

.59057 

.03896 

.62740 

.04240 

<§  47 

.51236 

.03254 

.55271 

.03570 

.59120 

.03901 

.62800 

.04246 

12     48 

.51305 

.03259 

.55336 

.03576 

.59183 

.03907 

.62860 

.04252 

i^  49 

.51374 

.03264 

.55402 

.03581 

.59245 

.03912 

.62919 

.04258 

^0  50 

8.51442 

0.03269 

8.55467 

0.03587 

8.59308 

0.03918 

8.62979 

0.04264 

24     51 

.51511 

.03274 

.55533 

.03592 

.59370 

.03924 

.63039 

.04270 

^5  52 

.51580 

.03279 

.55598 

.03597 

.59433 

.03929 

.63099 

.04276 

32     53 

.51648 

.03285 

.55664 

.03603 

.59495 

.03935 

.63159 

.04281 

5^  54 

.51717 

.03290 

.55729 

.03608 

.59558 

.03941 

.63218 

.04287 

40  55 

8.51785 

0.03295 

8.55794 

0.03614 

8.59620 

0.03946 

8.63278 

0.04293 

44     56 

.51854 

.03300 

.55859 

.03619 

.59682 

.03952 

.63338 

.04299 

45  57 

.51922 

.03305 

.55925 

.03624 

.59745 

.03958 

.63397 

.04305 

52     58 

.51990 

.03311 

.55990 

.03630 

.59807 

.03963 

.63457 

.04311 

5(?  59 

.52058 

.03316 

.56055 

.03635 

.59869 

.03969 

.63516 

.04317 

^0  60 

8.52127 

0.03321 

8.56120 

0.03641 

8.59931 

0.03975 

8.63576 

0.04323 

256 


Table  10.    Haversine  Table 


s 

^^  36-"^ 

24° 

1>'  40'" 

25° 

ih  44"^ 

26° 

Ih  48'" 

27° 

Hav. 

No. 

Hav. 

No. 

Hav. 

No. 

Hav. 

No. 

0       0 

8.63576 

0.04323 

8.67067 

0.04685 

8.70418 

0.05060 

8.73637 

0.05450 

4      1 

.63635 

.04329 

.67124 

.04691 

.70472 

.05067 

.73690 

.05456 

8       2 

.63695 

.04335 

.67181 

.04697 

.70527 

.05073 

.73742 

.05463 

12       3 

.63754 

.04340 

.67238 

.04703 

.70582 

.05079 

.73795 

.05470 

16       4 

.63813 

.04346 

.67295 

.04709 

.70636 

.05086 

.73847 

.05476 

20       5 

8.63872 

0.04352 

8.67352 

0.04715 

8.70691' 

0.05092 

8.73900 

0.05483 

24       6 

.63932 

.04358 

.67409 

.04722 

.70745 

.05099 

.73952 

.05489 

28       7 

.63991 

.04364 

.67465 

.04728 

.70800 

.05105 

.74005 

.05496 

5^   8 

.64050 

.04370 

.67522 

.04734 

.70854 

.05111 

.74057 

.05503 

5^   9 

.64109 

.04376 

.67579 

.04740 

.70909 

.05118 

.74109 

.05509 

40    10 

8.64168 

0.04382 

8.67635 

0.04746 

8.70963 

0.05124 

8.74162 

0.05516 

44   11 

.64227 

.04388 

.67692 

.04752 

.71017 

.05131 

.74214 

.05523 

48     12 

.64286 

.04394 

.67748 

.04759 

.71072 

.05137 

.74266 

.05529 

5;^  13 

.64345 

.04400 

.67805 

.04765 

.71126 

.05144 

.74318 

.05536 

5^  14 

.64404 

.04405 

.67861 

.04771 

.71180 

.05150 

.74371 

.05542 

s 

Ih  37'" 

24° 

Ih  U>n 

25° 

Ih  45'" 

26° 

Ih  49'" 

27° 

0     15 

8.64463 

0.04412 

8.67918 

0.04777 

8.71234 

0.05156 

8.74423 

0.05549 

4  16 

.64521 

.04418 

.67974 

.04783 

.71289 

.05163 

.74475 

.05556 

8     17 

.64580 

.04424 

.68030 

.04790 

.71343 

.05169 

.74527 

.05562 

i^  18 

.64639 

.04430 

.68087 

.04796 

.71397 

.05176 

.74579 

.05569 

16     19 

.64697 

.04436 

.68143 

.04802 

.71451 

.05182 

.74631 

.05576 

;gO  20 

8.64756 

0.04442 

8.68199 

0.04808 

8.71505 

0.05189 

8.74683 

0.05582 

^4  21 

.64815 

.04448 

.68256 

.04815 

.71559 

.05195 

.74735 

.05589 

^5  22 

.64873 

.04454 

.68312 

.04821 

.71613 

.05201 

.74787 

.05596 

5^  23 

.64932 

.04460 

.68368 

.04827 

.71667 

.05208 

.74839 

.05603 

36     24 

.64990 

.04466 

.68424 

.04833 

.71721 

.05214 

.74890 

.05609 

40  25 

8.65049 

0.04472 

8.68480 

0.04839 

8.71774 

0.05221 

S.74942 

0.05616 

44  26 

.65107 

.04478 

.68536 

.04846 

.71828 

.05227 

.74994 

.05623 

48     27 

.65165 

.04484 

.68592 

.04852 

.71882 

.05234 

.75046 

.05629 

5;^  28 

.65224 

.04490 

.68648 

.04858 

.71936 

.05240 

.75097 

.05636 

56     29 

.65282 

.04496 

.68704 

.04864 

.71989 

.05247 

.75149 

.05643 

s 

Ih  38'" 

24° 

Ih  42'" 

25° 

Ih  46'^ 

26° 

Ih  50"' 

27° 

0  30 

8.65340 

0.04502 

8.68760 

0.04871 

8.72043 

0.05253 

8.75201 

0.05649 

4  31 

.65398 

.04508 

.68815 

.04877 

.72097 

.05260 

.75252 

.05656 

^  32 

.65456 

.04514 

.68871 

.04883 

.72150 

.05266 

.75304 

.05663 

i^  33 

.65514 

.04520 

.68927 

.04890 

.72204 

.05273 

.75355 

.05670 

16     34 

.65572 

.04526 

.68983 

.04896 

.72257 

.05279 

.75407 

.05676 

^0  35 

8.65630 

0.04532 

8.69038 

0.04902 

8.72311 

0.05286 

8.75458 

0.05683 

24     36 

.65688 

.04538 

.69094 

.04908 

.72364 

.05292 

.75510 

.05690 

;g5  37 

.65746 

.04544 

.69149 

.04915 

.72418 

.05299 

.75561 

.05697 

5jg  38 

.65804 

.04550 

.69205 

.04921 

.72471 

.05305 

.75613 

.05703 

36     39 

.65862 

.04556 

.69260 

.04927 

.72525 

.05312 

.75664 

.05710 

40  40 

8.65920 

0.04562 

8.69316 

0.04934 

8.72578 

0.05318 

8.75715 

0.05717 

44    41 

.65978 

.04569 

.69371 

.04940 

.72631 

.05325 

.75767 

.05724 

45  42 

.66035 

.04575 

.69427 

.04946 

.72684 

.05331 

.75818 

.05730 

5:g  43 

.66093 

.04581 

.69482 

.04952 

.72738 

.05338 

.75869 

.05737 

56     44 

.66151 

.04587 

.69537 

.04959 

.72791 

.05345 

.75920 

.05744 

s   •' 

Ih  30'" 

24° 

Ih  43'" 

25° 

Ih  J^jm 

26° 

Ih   Jim 

27° 

0     45 

8.66208 

0.04593 

8.69593 

0.04965 

8.72844 

0.05351 

8.75972 

0.C5751 

4  46 

.66266 

.04599 

.69648 

.04971 

.72897 

.05358 

.76023 

.05757 

5  47 

.66323 

.04605 

.69703 

.04978 

.72950 

.05364 

.76074 

.05764 

12     48 

.66381 

.04611 

.69758 

.04984 

.73003 

.05371 

.76125 

.05771 

i^  49 

.66438 

.04617 

.69814 

.04990 

.73056 

.05377 

.76176 

.05778 

20     50 

8.66496 

0.04623 

8.69869 

0.04997 

8.73109 

0.05384 

8.76227 

0.05785 

;?4  51 

.66553 

.04629 

.69924 

.05003 

.73162 

.05390 

.76278 

.05791 

:^5  52 

.66610 

.04636 

.69979 

.05009 

.73215 

.05397 

.76329 

.05798 

32     53 

.66668 

.04642 

.70034 

.05016 

.73268 

.05404 

.76380 

.05805 

5^  54 

.66725 

.04648 

.70089 

.05022 

.73321 

.05410 

.76431 

.05812 

40     55 

8.66782 

0.04654 

8.70144 

0.05028 

8.73374 

0.05417 

8.76481 

0.05819 

44  56 

.66839 

.04660 

.70198 

.05035 

.73426 

.05423 

.76532 

.05825 

45  57 

.66896 

.04666 

.70253 

.05041 

.73479 

.05430 

.76583 

.05832 

52     58 

.66953 

.04672 

.70308 

.05048 

.73532 

.05436 

.76634 

.05839 

5^  59 

.67010 

.04678 

.70363 

.05054 

.73584 

.05443 

.76684 

.05846 

60     60 

8.67067 

0.04685 

8.70418 

0.05060 

8.73637 

0.05450 

8.76735 

0.05853 

Table  10.    Haversine  Table 


267 


s 

Ih  52^ 

28° 

Jh  5S>n 

29° 

2h  0'« 

30° 

2h  ^m 

31° 

Hav. 

No. 

Hav. 

1    x\0. 

Hav. 

No. 

Hav. 

No. 

0       0 

8.76735 

0.05853 

8.79720 

0.06269 

8.82599 

0.06699 

8.85380 

0.07142 

4      1 

.76786 

.05859 

.79769 

.06276 

.82646 

.06706 

.85425 

.07149 

8      2 

.76836 

.05866 

.79818 

.06283 

.82694 

.06713 

.85471 

.07157 

12       3 

.76887 

.05873 

.79866 

.06290 

.82741 

.06721 

.85516 

.07164 

16       4 

.76938 

.05880 

.79915 

.06297 

.82788 

.06728 

.85562 

.07172 

20       5 

8.76988 

0.05887 

8.79964 

0.06304 

8.82835 

0.06735 

<S.  85607 

0.07179 

24       6 

.77039 

.05894 

.80013 

.06311 

.82882 

.06742 

.85653 

.07187 

28       7 

.77089 

.05901 

.80061 

.06318 

.82929 

.06750 

.85698 

.07194 

S;g   8 

.77139 

.05907 

.80110 

.06326 

.82976 

.06757 

.85743 

.07202 

56   9 

.77190 

.05914 

.80158 

.06333 

.83023 

.06764 

.85789 

.07209 

40     10 

8.77240 

0.05921 

8.80207 

0.06340 

8.83069 

0.06772 

8.85834 

0.07217 

44  11 

.77291 

.05928 

.80256 

.06347 

.83116 

.06779 

.85879 

.07224 

48    12 

.77341 

.05935 

.80304 

.06354 

.83163 

.06786 

.85925 

.07232 

J^  13 

.77391 

.05942 

.80353 

.06361 

.83210 

.06794 

.85970 

.07239 

56  14 

.77441 

.05949 

.80401 

.06368 

.83257 

.06801 

.86015 

.07247 

s 

Ih  ^s'" 

28° 

Ih  57'" 

29° 

2h   /'" 

30° 

2h  5'"^ 

31° 

0     15 

8.77492 

0.05955 

8.80449 

0.06375 

8.83303 

0.06808 

8.86060 

0.07254 

4  16 

.77542 

.05962 

.80498 

.06382 

.83350 

.06816 

.86105 

.07262 

8     17 

.77592 

.05969 

.80546 

.06389 

.8.3397 

.06823 

.86151 

.07270 

i^  18 

.77642 

.05976 

.80595 

.06397 

.83444 

.06830 

.86196 

.07277 

16     19 

.77692 

.05983 

.80643 

.06404 

.83490 

.06838 

.86241 

.07285 

^0  20 

8.77742 

0.05990 

8.80691 

0.06411 

8.83537 

0.06845 

8.86286 

0.07292 

^4  21 

.77792 

.05997 

.80739 

.06418 

.83583 

.06852 

.86331 

.07300 

28     22 

.77842 

.06004 

.80788 

.06425 

.83630 

.06860 

.86376 

.07307 

5^  23 

.77892 

.06011 

.80836 

.06432 

.83676 

.06867 

.86421 

.07315 

36     24 

.77942 

.06018 

.80884 

.06439 

.83723 

.06874 

.86466 

.07322 

40  25 

8.77992 

0.06024 

8.80932 

0.06446 

8.83769 

0.06882 

8.86511 

0.07330 

44  26 

.78042 

.06031 

.80980 

.06454 

.83816 

.06889 

.86556 

.07338 

48     27 

.78092 

.06038 

.81028 

.06461 

.83862 

.06896 

.86600 

.07345 

Ji?  28 

.78142 

.06045 

.81076 

.06468 

.83909 

.06904 

.86645 

.07353 

56     29 

.78191 

.06052 

.81124 

.06475 

.83955 

.06911 

.86690 

.07360 

s 

Ih  54'" 

28° 

Ih  s8"^ 

29° 

2h  2'"' 

30° 

2h  6'" 

31° 

0  30 

8.78241 

0.06059 

8.81172 

0.06482 

8.84002 

0.06919 

8.86735 

0.07368 

4     31 

.78291 

.06066 

.81220 

.06489 

.84048 

.06926 

.86780 

.07376 

5  32 

.78341 

.06073 

.81268 

.06497 

.84094 

.06933 

.86825 

.07383 

i^  33 

.78390 

.06080 

.81316 

.06504 

.84140 

.06941 

.86869 

.07391 

16     34 

.78440 

.06087 

.81364 

.06511 

.84187 

.06948 

.86914 

.07398 

;gO  35 

8.78490 

0.06094 

8.81412 

0.06518 

8.842.33 

0.06956 

8.86959 

0.07406 

24     36 

.78539 

.06101 

.81460 

.06525 

.84279 

.06963 

.87003 

.07414 

^5  37 

.78589 

.06108 

.81508 

.06532 

.84325 

.06970 

.87048 

.07421 

S^  38 

.78638 

.06115 

.81555 

.06540 

.84371 

.06978 

.87093 

.07429 

36     39 

.78688 

.06122 

.81603 

.06547 

.84417 

.06985 

.87137 

.07437 

40  40 

8.78737 

0.06129 

8.81651 

0.06554 

8.84464 

0.06993 

8.87182 

0.07444 

44    41 

.78787 

.06136 

.81699 

.06561 

.84510 

.07000 

.87226 

.07452 

45  42 

.78836 

.06143 

.81746 

.06568 

.84556 

.07007 

.87271 

.07459 

5^  43 

.78885 

.06150 

.81794 

.06576 

.84602 

.07015 

.87315 

.07467 

56     44 

.78935 

.06157 

.81841 

.06583 

.84648 

07022 

.87360 

.07475 

s 

Ih  55"" 

28° 

Ih  50"" 

29° 

2h  3'" 

30° 

^h   7''i 

31° 

0  45 

8.78984 

0.06164 

8.81889 

0.06590 

8.84694 

0.07030 

8.87404 

0.07482 

4     46 

.79033 

.06171 

.81937 

.06597 

.84740 

.07037 

.87448 

.07490 

<§  47 

.7.9082 

.06178 

.81984 

.06605 

.84785 

.07045 

.87493 

.07498 

12     48 

.79132 

.06185 

.82032 

.06612 

.84831 

.07052 

.87537 

.07505 

M  49 

.79181 

.06192 

.82079 

.06619 

.84877 

.07059 

.87582 

.07513 

:gO  50 

8.79230 

0.06199 

8.82126 

0.06626 

8.84923 

0.07067 

8.87626 

0.07521 

24     51 

.79279 

.06206 

.82174 

.06633 

.84969 

.07074 

.87670 

.07528 

^5  52 

.79328 

.06213 

.82221 

.06641 

.85015 

.07082 

.87714 

.07536 

32     53 

.79377 

.06220 

.82269 

.06648 

.85060 

.07089 

.87759 

.07544 

56  54 

.79426 

.06227 

.82316 

.06655 

.85106 

.07097 

.87803 

.07551 

40  55 

8.79475 

0.06234 

8.82363 

0.06662 

8.85152 

0.07104 

8.87847 

0.07559 

44     56 

.79524 

.06241 

.82410 

.06670 

.85197 

.07112 

.87891 

.07567 

45  57 

.79573 

.06248 

.82458 

.06677 

.85243 

.07119 

.87935 

.07574 

52     58 

.79622 

.06255 

.82505 

.06684 

.85289 

.07127 

.87980 

.07582 

56  59 

.79671 

.06262 

.82552 

.06691 

.85334 

.07134 

.88024 

.07590 

60  60 

8.79720 

0.06269 

8.82599 

0.06699 

8.85380 

0.07142 

8.88068 

0.07598 

258 


Table  10.    Haversiue  Table 


s 

2h  gm 

32° 

2h  23'" 

33° 

2h   Jfjm 

34° 

2h  20"" 

35° 

Hav. 

No. 

Hav. 

8.90668 

No. 

Hav. 

8.93187 

No. 

Hav. 

8.95628 

No. 

0 

0 

8.88068 

0.07598 

0.08066 

0.08548 

0.09042 

4 

1 

.88112 

.07605 

.90711 

.08074 

.93228 

.08556 

.95668 

.09051 

8 

2 

.88156 

.07613 

.90754 

.08082 

.93270 

.08564 

.95709 

.09059 

12 

3 

.88200 

.07621 

.90796 

.08090 

.93311 

.08573 

.95749 

.09067 

16 

4 

.88244 

.07628 

.90839 

.08098 

.93352 

.08581 

.95789 

.09076 

20 

5 

8.88288 

0.07636 

8.90881 

0.08106 

8.93393 

0.08589 

8.95828 

0.09084 

24 

6 

.88332 

.07644 

.90924 

.08114 

.93435 

.08597 

.95868 

.09093 

28 

7 

.88375 

.07652 

.90966 

.08122 

.93476 

.08605 

.95908 

.09101 

32 

8 

.88419 

.07659 

.91009 

.08130 

.93517 

.08613 

.95948 

.09109 

36 

9 

.88463 

.07667 

.91051 

.08138 

.93558 

.08621 

.95988 

.09118 

40 

10 

8.88507 

0.07675 

8.91094 

0.08146 

8.93599 

0.08630 

8.96028 

0.09126 

44 

11 

.88551 

.07683 

.91136 

.08154 

.93640 

.08638 

.96068 

.09134 

48 

12 

.88595 

.07690 

.91179 

.08162 

.93681 

.08646 

.96108 

.09143 

52 

13 

.88638 

.07698 

.91221 

.08170 

.93722 

.08654 

.96148 

.09151 

56 

14 

.88682 

.07706 

.91263 

.08178 

.93764 

.08662 

.96187 

.09160 

s 

I 

2h  gm 

32° 

Oh   Ijm 

33° 

2h  17m 

34° 

2h  21"^ 

35° 

0 

Ts" 

8.88726 

0.07714 

8.91306 

0.08186 

8.93805 

0.08671 

8.96227 

0.09168 

4 

16 

.88769 

.07721 

.91348 

.08194 

.93846 

.08679 

.96267 

.09176 

8 

17 

.88813 

.07729 

.91390 

.08202 

.93886 

.08687 

.96307 

.09185 

12 

18 

.88857 

.07737 

.91432 

.08210 

.93927 

.08695 

.96346 

.09193 

16 

19 

.88900 

.07745 

.91475 

.08218 

.93968 

.08703 

.96386 

.09202 

20 

20 

8.88944 

0.07752 

8.91517 

0.08226 

8.94009 

0.08711 

8.96426 

0.09210 

24 

21 

.88988 

.07760 

.91559 

.08234 

.94050 

.08720 

.96465 

.09218 

28 

22 

.89031 

.07768 

.91601 

.08242 

.94091 

.08728 

.96505 

.09227 

32 

23 

.89075 

.07776 

.91643 

.08250 

.94132 

.08736 

.96545 

.09235 

36 

24 

.89118 

.07784 

.91685 

.08258 

.94173 

.08744 

.96584 

.09244 

40 

25 

8.89162 

0.07791 

8.91728 

0.08266 

8.94213 

0.08753 

8.96624 

0.09252 

44 

26 

.89205 

.07799 

.91770 

.08274 

.94254 

.08761 

.96663 

.09260 

48 

27 

.89248 

.07807 

.91812 

.08282 

.94295 

.08769 

.96703 

.09269 

62 

28 

.89292 

.07815 

.91854 

.08290 

.94336 

.08777 

.96742 

.09277 

56 

s 

29 

.89335 

.07823 

.91896 

.08298 

.94376 

.08785 

.96782 

.09286 

2h  10"" 

32" 

2h  l^m 

33° 

2h  IS"" 

34° 

2h  22'" 

35° 

0 

30 

8.89379 

0.07830 

8.91938 

0.08306 

8.94417 

0.08794 

8.96821 

0.09294 

4 

31 

.89422 

.07838 

.91980 

.08314 

.94458 

.08802 

.96861 

.09303 

8 

32 

.89465 

.07846 

.92022 

.08322 

.94498 

.08810 

.96900 

.09311 

12 

33 

.89509 

.07854 

.92064 

.08330 

.94539 

.08818 

.96940 

.09320 

16 

34 

.89552 

.07862 

.92105 

.08338 

.94580 

.08827 

.96979 

.09328 

20 

35 

8.89595 

0.07870 

8.92147 

0.08346 

8.94620 

0.08835 

8.97018 

0.09337 

24 

36 

.89638 

.07877 

.92189 

.08354 

.94661 

.08843 

.97058 

.09345 

28 

37 

.89681 

.07885 

.92231 

.08362 

.94701 

.08851 

.97097 

.09353 

32 

38 

.89725 

.07893 

.92273 

.08370 

.94742 

.08860 

.971.36 

.09362 

36 

39 

.89768 

.07901 

.92315 

.08378 

.94782 

.08868 

.97176 

.09370 

40 

40 

8.89811 

0.07909 

8.92356 

0.08386 

8.94823 

0.08876 

8.97215 

0.09379 

44 

41 

.89854 

.07917 

.92398 

.08394 

.94863 

.08885 

.97254 

.09387 

48 

42 

.89897 

.07924 

.92440 

.08402 

.94904 

.08893 

.97294 

.09396 

52 

43 

.89940 

.07932 

.92482 

.08410 

.94944 

.08901 

.97333 

.09404 

56 

44 

.89983 

.07940 

.92523 

.08418 

.94985 

.08909 

.97372 

.09413 

s 

0h  Jjm 

32° 

2h  15m 

33° 

2h  J(J'" 

34° 

2h  23"" 

35° 

0 

45 

8.90026 

0.07948 

8.92565 

0.08427 

8.95025 

0.08918 

8.97411 

0.09421 

4 

46 

.90069 

.07956 

.92607 

.08435 

.95065 

.08926 

.97450 

.09430 

8 

47 

.90112 

.07964 

.92648 

.08443 

.95106 

.08934 

.97489 

.09438 

12 

48 

.90155 

.07972 

.92690 

.08451 

.95146 

.08943 

.97529 

.09447 

16 

49 

.90198 

.07980 

.92731 

.08459 

.95186 

.08951 

.97568 

.09455 

20 

50 

8.90241 

0.07987 

8.92773 

0.08467 

8.95227 

0.08959 

8.97607 

0.09464 

24 

51 

.90284 

.07995 

.92814 

.08475 

.95267 

.08967 

.97646 

.09472 

28 

52 

.90326 

.08003 

.92856 

.08483 

.95307 

.08976 

.97685 

.09481 

32 

53 

.90369 

.08011 

.92897 

.08491 

.95347 

.08984 

.97724 

.09489 

36 

54 

.90412 

.08019 

.92939 

.08499 

.95388 

.08992 

.97763 

.09498 

40 

55 

8.90455 

0.08027 

8.92980 

0.08508 

8.95428 

0.09001 

8.97802 

0.09506 

44 

56 

.90498 

.08035 

.93022 

.08516 

.95468* 

09009 

.97841 

.09515 

48 

57 

.90540 

.08043 

.93063 

.08524 

.95508 

.09017 

.97880 

.09524 

52 

58 

.90583 

.08051 

.93104 

.08532 

.95548 

.09026 

.97919 

.09532 

56 

59 

.90626 

.08059 

.93146 

.08540 

.95588 

.09034 

.97958 

.09541 

60 

60 

8.90668 

0.08066 

8.93187 

0.08548 

8.95628 

0.09042 

8.97997 

0.09549 

Table  10.    Hayersine  Table 


259 


s 

2h  24."' 

36° 

2h  28"^ 

37° 

2h  32"' 

38° 

2h  36m    . 

39° 

Hav. 

No. 

Hav. 

No. 

Hav. 

No. 

Hav. 

No. 

0 

0 

8.97997 

0.09549 

9.00295 

0.10068 

9.02528 

0.10599 

9.04699 

0.11143 

4 

1 

.98035 

.09558 

.00333 

.10077 

.02565 

.10608 

.04735 

.11152 

8 

2 

.98074 

.09566 

.00371 

.10086 

.02602 

.10617 

.04770 

.11161 

12 

3 

.98113 

.09575 

.00408 

.10095 

.02638 

.10626 

.04806 

.11170 

16 

4 

.98152 

.09583 

.00446 

.10103 

.02675 

.10635 

.04842 

.11179 

20 

5 

8.98191 

0.09592 

9.00484 

0.10112 

9.02712 

0.10644 

9.04877 

0.11189 

24 

6 

.98229 

.09601 

.00522 

.10121 

.02748 

.10653 

.04913 

.11198 

28 

7 

.98268 

.09609 

.00559 

.10130 

.02785 

.10662 

.04948 

.11207 

32 

8 

.98307 

.09618 

.00597 

.10138 

.02821 

.10671 

.04984 

.11216 

36 

9 

.98346 

.09626 

.00634 

.10147 

.02858 

.10680 

.05019 

.11225 

40 

10 

8.98384 

0.09635 

9.00672 

0.10156 

9.02894 

0.10689 

9.05055 

0.11234 

44 

11 

.98423 

.09643 

.00710 

.10165 

.02931 

.10698 

.05090 

.11244 

48 

12 

.98462 

.09652 

.00747 

.10174 

.02967 

.10707 

.05126 

.11253 

52 

13 

.98500 

.09661 

.00785 

.10182 

.03004 

.10716 

.05161 

.11262 

56 

14 

.98539 

.09669 

.00822 

.10191 

.03040 

.10725 

.05197 

.11271 

s 

2h  25"' 

36° 

2h  29"' 

37° 

2h  33m 

38° 

2h  37m 

39° 

0 

15 

8.98578 

0.09678 

9.00860 

0.10200 

9.03077 

0.10734 

9.05232 

0.11280 

4 

16 

.98616 

.09686 

.00897 

.10209 

.03113 

.10743 

.05268 

.11290 

8 

17 

.98655 

.09695 

.00935 

.10218 

.03150 

.10752 

.05303 

.11299 

12 

18 

.98693 

.09704 

.00972 

.10226 

.03186 

.10761 

.05339 

.11308 

16 

19 

.98732 

.09712 

.01009 

.10235 

.03222 

.10770 

.05374 

.11317 

20 

20 

8.98770 

0.09721 

9.01047 

0.10244 

9.03259 

0.10779 

9.05409 

0.11326 

24 

21 

.98809 

.09729 

.01084 

.10253 

.03295 

.10788 

.05445 

.11336 

28 

22 

.98847 

.09738 

.01122 

.10262 

.03331 

.10797 

.05480 

.11345 

32 

23 

.98886 

.09747 

.01159 

.10270 

.03368 

.10806 

.05515 

.11354 

36 

24 

.98924 

.09755 

.01196 

.10279 

.03404 

.10815 

.05551 

.11363 

40 

25 

8.98963 

0.09764 

9.01234 

0.10288 

9.03440 

0.10824 

9.05586 

0.11373 

44 

26 

.99001 

.09773 

.01271 

.10297 

.03476 

.10833 

.05621 

.11382 

48 

27 

.99039 

.09781 

.01308 

.10306 

.03513 

.10842 

.05656 

.11391 

52 

28 

.99078 

.09790 

.01345 

.10315 

.03549 

.10851 

.05692 

.11400 

56 
s 

29 

.99116 

.09799 

.01383 

.10323 

.03585 

.10861 

.05727 

.11410 

2h  26^ 

36° 

2h  30^ 

37° 

Oh  3/m 

38° 

2h  3S"^ 

39° 

0 

30 

8.99154 

0.09807 

9.01420 

0.10332 

9.03621 

0.10870 

9.05762 

0.11419 

4 

31 

.99193 

.09816 

.01457 

.10341 

.03657 

.10879 

.05797 

.11428 

8 

32 

.99231 

.09824 

.01494 

.10350 

.03694 

.10888 

.05832 

.11437 

12 

33 

.99269 

.09833 

.01531 

.10359 

.03730 

.10897 

.05867 

.11447 

16 

34 

.99307 

.09842 

.01569 

.10368 

.03766 

.10906 

.05903 

.11456 

20 

35 

8.99346 

0.09850 

9.01606 

0.10377 

9.03802 

0.10915 

9.05938 

0.11465 

24 

36 

.993X4 

.09859 

.01643 

.10386 

.03838 

.10924 

.05973 

.11474 

28 

37 

.99422 

.09868 

.01680 

.10394 

.03874 

.10933 

.06008 

.11484 

32 

38 

.99460 

.09876 

.01717 

.10403 

.03910 

.10942 

.06043 

.11493 

36 

39 

.99498 

.09885 

.01754 

.10412 

.03946 

.10951 

.06078 

.11502 

40 

40 

8.99536 

0.09894 

9.01791 

0.10421 

9.03982 

0.10960 

9.06113 

0.11511 

44 

41 

.99575 

.09903 

.01828 

.10430 

.04018 

.10969 

.06148 

.11521 

48 

42 

.99613 

.09911 

.01865 

.10439 

.04054 

.10978 

.06183 

.11530 

52 

43 

.99651 

.09920 

.01902 

.10448 

.04090 

.10988 

.06218 

.11539 

56 

44 

.99689 

.09929 

.01939 

.10457 

.04126 

.10997 

.06253 

.11549 

s        ' 

2h  27'" 

36° 

2h  3Jm 

37° 

2h  5.5m 

38° 

2h  39m 

39° 

0 

45 

8.99727 

0.09937 

9.01976 

0.10466 

9.04162 

0.11006 

9.06288 

0.11558 

4 

46 

.99765 

.09946 

.02013 

.10474 

.04198 

.11015 

.06323 

.11567 

8 

47 

.99803 

.09955 

.02050 

.10483 

.04234 

.11024 

.06358 

.11577 

12 

48 

.99841 

.09963 

.02087 

.10492 

.04270 

.11033 

.06393 

.11586 

16 

49 

.99879 

.09972 

.02124 

.10501 

.04306 

.11042 

.06428 

.11595 

20 

50 

8.99917 

0.09981 

9.02161 

0.10510 

9.04341 

0.11051 

9.06462 

0.11604 

24 

51 

.99955 

.09990 

.02197 

.10519 

.04377 

.11060 

.06497 

.11614 

28 

52 

.99993 

.09998 

.02234 

.10528 

.04413 

.11070 

.06532 

.11623 

32 

53 

9.00031 

.10007 

.02271 

.10537 

.04449 

.11079 

.06567 

.11632 

36 

54 

.00068 

.10016 

.02308 

.10546 

.04485 

.11088 

.06602 

.11642 

40 

55 

9.00106 

0.10025 

9.02345 

0.10555 

9.04520 

0.11097 

9.06637 

0.11651 

44 

56 

.00144 

.10033 

.02381 

.10564 

.04556 

.11106 

.06671 

.11660 

48 

57 

.00182 

.10042 

.02418 

.10573 

.04592 

.11115 

.06706 

.11670 

52 

58 

.00220 

.10051 

.02455 

.10582 

.04628 

.11124 

.06741 

.11679 

56 

59 

.00258 

.10059 

.02492 

.10591 

.04663 

.11134 

.06776 

.11688 

60 

60 

9.00295 

0.10068 

9.02528 

0.10599 

9.04699 

0.11143 

9.06810 

0.11698 

260 


Table  10.    Haversine  Table 


s 

Hav. 

9.06810 

40° 
No. 

2h  44"" 

41° 

2^  48^ 

42° 

2h  52^ 

43° 

Hav. 

No. 
0.12265 

Hav. 

No. 

Hav. 

No. 

0       0 

0.11698 

9.08865 

9.10866 

0.12843 

9.12815 

0.13432 

4      1 

.06845 

.11707 

.08899 

.12274 

.10899 

.12852 

.12847 

.13442 

8       2 

.06880 

.11716 

.08933 

.12284 

.10932 

.12862 

.12879 

.13452 

12       3 

.06914 

.11726 

.08960 

.12293 

.10965 

.12872 

.12911 

.13462 

16       4 

.06949 

.11735 

.09000 

.12303 

.10997 

.12882 

.12943 

.13472 

20       5 

9.06984 

0.11745 

,1.09034 

0.12312 

9.11030 

0.12891 

9.12975 

0.13482 

24       6 

.07018 

.11754 

.09008 

.12322 

.11063 

.12901 

.13007 

.13492 

28       7 

.07053 

.11763 

.09101 

.12331 

.11096 

.12911 

.13039 

.13502 

5^   8 

.07088 

.11773 

.09135 

.12341 

.11129 

.12921 

.13071 

.13512 

55   9 

.07122 

.11782 

.09169 

.12351 

.11161 

.12930 

.13103 

.13522 

40     10 

9.07157 

0.11791 

9.09202 

0.12360 

9.11194 

0.12940 

9.13135 

0.13532 

44  11 

.07191 

.11801 

.09236 

.12370 

.11227 

.12950 

.13167 

.13542 

48     12 

.07226 

.11810 

.09269 

.12379 

.11260 

.12960 

.13199 

.13552 

5^  13 

.07260 

.11820 

.09303 

.12389 

.11292 

.12970 

.13231 

.13562 

56)  14 

.07295 

.11829 

.09337 

.12398 

.11325 

.12979 

.13263 

.13571 

s 

2h  41^ 

40° 

2h  45-^ 

41° 

2h  j^gm 

42° 

2*^  53'" 

43° 

0     15 

9.07329 

0.11838 

9.09370 

0.12408 

9.11358 

0.12989 

9.13295 

0.13581 

4  16 

.07364 

.11848 

.09404 

.12418 

.11391 

.12999 

.13326 

.13591 

8     17 

.07398 

.11857 

.09437 

.12427 

.11423 

.13009 

.13358 

.13601 

i^  18 

.07433 

.11867 

.09471 

.12437 

.11456 

.13018 

.13390 

.13611 

16     19 

.07467 

.11876 

.09504 

.12446 

.11489 

.13028 

.13422 

.13621 

^0  20 

9.07501 

0.11885 

9.09538 

0.12456 

9.11521 

0.13038 

9.13454 

0.13631 

^4  21 

.07536 

.11895 

.09571 

.12466 

.11554 

.13048 

.13486 

.13641 

^5  22 

.07570 

.11904 

.09605 

.12475 

.11586 

.13058 

.13517 

.13651 

3^  23 

.07605 

.11914 

.09638 

.12485 

.11619 

.13067 

.13549 

.13661 

36     24 

.07639 

.11923 

.09672 

.12494 

.11652 

.13077 

.13581 

.13671 

40  25 

9.07673 

0.11933 

9.09705 

0.12504 

9.11084 

0.13087 

9.13613 

0.13681 

44  26 

.07708 

.11942 

.09739 

.12514 

.11717 

.13097 

.13644 

.13691 

48     27 

.07742 

.11951 

.09772 

.12523 

.11749 

.13107 

.13676 

.13701 

5;g  28 

.07776 

.11961 

.09805 

.12533 

.11782 

.13116 

.13708 

.13711 

56     29 

.07810 

.11970 

.09839 

.12543 

.11814 

.13126 

.13739 

.13721 

s   ' 

2h  42^ 

40^ 

2h  J^(jm 

41° 

2^50"" 

42° 

2^^  54"" 

43° 

0  30 

9.07845 

0.11980 

9.09872 

0.12552 

9.11847 

0.13136 

9.13771 

0.13731 

4  31 

.07879 

.11989 

.09905 

.12562 

.11879 

.13146 

.13803 

.13741 

S  32 

.07913 

.11999 

.09939 

.12572 

.11912 

.13156 

.13834 

.13751 

12     33 

.07947 

.12008 

.09972 

.12581 

.11944 

.13166 

.13866 

.13761 

i^  34 

.07981 

.12018 

.10005 

.12591 

.11977 

.13175 

.13898 

.13771 

^0  35 

9.08016 

0.12027 

9.10039 

0.12600 

9.12009 

0.13185 

9.13929 

0.13781 

24     36 

.08050 

.12036 

.10072 

.12610 

.12041 

.13195 

.13961 

.13791 

^5  37 

.08084 

.12046 

.10105 

.12620 

.12074 

.13205 

.13992 

.13801 

32     38 

.08118 

.12055 

.10138 

.12629 

.12106 

.13215 

.14024 

.13811 

35  39 

.08152 

.12065 

.10172 

.12639 

.12139 

.13225 

.14056 

.13822 

40  40 

9.08186 

0.12074 

9.10205 

0.12649 

9.12171 

0.13235 

9.14087 

0.13832 

44    41 

.08220 

.12084 

.10238 

.12658 

.12203 

.13244 

.14119 

.13842 

45  42 

.08254 

.12093 

.10271 

.12668 

.12236 

.13254 

.14150 

.13852 

52     43 

.08288 

.12103 

.10304 

.12678 

.12268 

.13264 

.14182 

.13862 

56  44 

.08323 

.12112 

.10337 

.12687 

.12300 

.13274 

.14213 

.13872 

s 

2h  ^3»i 

40° 

2h  J^jm 

41° 

2h  gjm 

42° 

2h   Jjm 

43° 

0  45 

9.08357 

0.12122 

9.10371 

0.12697 

9.12332 

0.13284 

9.14245 

0.13882 

4  46 

.08391 

.12131 

.10404 

.12707 

.12365 

.13294 

.14276 

.13892 

5  47 

.08425 

.12141 

.10437 

.12717 

.12397 

.13304 

.14307 

.13902 

12     48 

.08459 

.12150 

.10470 

.12726 

.12429 

.13314 

.14339 

.13912 

i5  49 

.08492 

.12160 

.10503 

.12736 

.12461 

.13323 

.14370 

.13922 

20     50 

9.08526 

0.12169 

9.10536 

0.12746 

9.12494 

0.13333 

9.14402 

0.13932 

^4  51 

.08560 

.12179 

.10569 

.12755 

.12526 

.13343 

.14433 

.13942 

^5  52 

.08594 

.12188 

.10602 

.12765 

.12558 

.13353 

.14465 

.13952 

32     53 

.08628 

.12198 

.10635 

.12775 

.12590 

.13363 

.14496 

.13962 

35  54 

.08662 

.12207 

.10668 

.12784 

.12622 

.13373 

.14527 

.13972 

40     55 

9.08696 

0.12217 

9.10701 

0.12794 

9.12055 

0.13383 

9.14559 

0.13983 

44  56 

.08730 

.12226 

.10734 

.12804 

.12687 

.13393 

.14590 

.13993 

45  57 

.08764 

.12236 

.10767 

.12814 

.12719 

.13403 

.14621 

.14003 

52     58 

.08797 

.12245 

.10800 

.12823 

.12751 

.13412 

.14653 

.14013 

55  59 

.08831 

.12255 

.10833 

.12833 

.12783 

.13422 

.14084 

.14023 

60     60 

9.08865 

0.12265 

9.10866 

0.12843 

9.12815 

0.13432 

9.14715 

0.14033 

Table  10.    Haversine  Table 


261 


s 

2h  ,^0>n 

44° 

Qh  o'« 

45° 

gh  J^m 

Hav. 

46° 
No. 

gh  gm 

47° 

Hav. 

No. 

Hav. 

No. 

Hav. 

9.20140 

No. 

0       0 

9.14715 

0.14033 

9.16568 

0.14645 

9.18376 

0.15267 

0.15900 

4      1 

.14746 

.14043 

.16598 

.14655 

.18405 

.15278 

.20169 

.15911 

8       2 

.14778 

.14053 

.16629 

.14665 

.18435 

.15288 

.20198 

.15921 

12       3 

.14809 

.14063 

.16659 

.14676 

.18465 

.15298 

.20227 

.15932 

16       4 

.14840 

.14073 

.16690 

.14686 

.18495 

.15309 

.20256 

.15943 

20       5 

9.14871 

0.14084 

9.16720 

0.14696 

9.18524 

0.15319 

9.20285 

0.15953 

24       6 

.14902 

.14094 

.16751 

.14706 

.18554 

.15330 

.20314 

.15964 

28       7 

.14934 

.14104 

.16781 

.14717 

.18584 

.15340 

.20343 

.15975 

32       8 

.14965 

.14114 

.16812 

.14727 

.18613 

.15351 

.20372 

.15985 

5&   9 

.14996 

.14124 

.16842 

.14737 

.18643 

.15361 

.20401 

.15996 

40     10 

9.15027 

0.14134 

9.16872 

0.14748 

9.18673 

0.15372 

9.20430 

0.16007 

44  11 

.15058 

.14144 

.16903 

.14758 

.18702 

.15382 

.20459 

.16017 

48     12 

.15089 

.14154 

.16933 

.14768 

.18732 

.15393 

.20488 

.16028 

5^  13 

.15120 

.14165 

.16963 

.14779 

.18762 

.15403 

.20517 

.16039 

,5^  14 

.15152 

.14175 

.16994 

.14789 

.18791 

.15414 

.20546 

.16049 

s 

2h  gjm 

44° 

Qh  pn 

45° 

gh  5»i 

46° 

gh  gm 

47° 

0     15 

9.15183 

0.14185 

9.17024 

0.14799 

9.18821 

0.15424 

9.20574 

0.16060 

4  16 

.15214 

.14195 

.17054 

.14810 

.18850 

.15435 

.20603 

.16071 

8     17 

.15245 

.14205 

.17085 

.14820 

.18880 

.15445 

.20632 

.16081 

i^  18 

.15276 

.14215 

.17115 

.14830 

.18909 

.15456 

.20661 

.16092 

16     19 

.15307 

.14226 

.17145 

.14841 

.18939 

.15466 

.20690 

.16103 

^0  20 

9.15338 

0.14236 

9.17175 

0.14851 

9.18968 

0.15477 

9.20719 

0.16113 

;^4  21 

.15369 

.14246 

.17206 

.14861 

.18998 

.15487 

.20748 

.16124 

:?<?  22 

.15400 

.14256 

.17236 

.14872 

.19027 

.15498 

.20770 

.16135 

5^  23 

.15431 

.14266 

.17266 

.14882 

.19057 

.15509 

.20805 

.16146 

36    24 

.15462 

.14276 

.17296 

.14892 

.19086 

.15519 

.20834 

.16156 

40  25 

9.15493 

0.14287 

9.17327 

0.14903 

9.19116 

0.15530 

9.20863 

0.16167 

44  26 

.15524 

.14297 

.17357 

.14913 

.19145 

.15540 

.20891 

.16178 

48     27 

.15555 

.14307 

.17387 

.14923 

.19175 

.15551 

.20920 

.16188 

5^  28 

.15585 

.14317 

.17417 

.14934 

.19204 

.15561 

.20949 

.16199 

56     29 

.15616 

.14327 

.17447 

.14944 

.19234 

.15572 

.20978 

.16210 

s 

2h   5^m 

440 

gh  2"^ 

45° 

gh  Qm 

46° 

gh  10"" 
9.21006 

47° 
0.16220 

0  30 

9.15647 

0.14337 

9.17477 

0.14955 

9.19263 

0.15582 

4  31 

.15678 

.14348 

.17507 

.14965 

.19292 

.15593 

.21035 

.16231 

5  32 

.15709 

.14358 

.17538 

.14975 

.19322 

.15603 

.21064 

.16242 

1;^  33 

.15740 

.14368 

.17568 

.14986 

.19351 

.15614 

.21092 

.16253 

16     34 

.15771 

.14378 

.17598 

.14996 

.19381 

.15625 

.21121 

.16263 

^0  35 

9.15802 

0.14388 

9.17628 

0.15006 

9.19410 

0.15635 

9.21150 

0.16274 

24     36 

.15832 

.14399 

.17658 

.15017 

.19439 

.15646 

.21178 

.16285 

•^5  37 

.15863 

.14409 

.17688 

.15027 

.19469 

.15656 

.21207 

.16296 

5^  38 

.15894 

.14419 

.17718 

.15038 

.19498 

.15667 

.21236 

.16306 

36     39 

.15925 

.14429 

.17748 

.15048 

.19527 

.15677 

.21264 

.16317 

40  40 

9.15955 

0.14440 

9.17778 

0.15058 

9.19557 

0.15688 

9.21293 

0.16328 

44    41 

.15986 

.14450 

.17808 

.15069 

.19586 

.15699 

.21322 

.16339 

45  42 

.16017 

.14460 

.17838 

.15079 

.19615 

.15709 

.21350 

.16349 

5^  43 

.16048 

.14470 

.17868 

.15090 

.19644 

.15720 

.21379 

.16360 

56     44 

.16078 

.14480 

.17898 

.15100 

.19674 

.15730 

.21407 

gh  Jjm 

.16371 
47° 

s 

2h  SQm 

44° 

gh  s"" 

45° 

gh  ^m 

46° 

0  45 

9.16109 

0.14491 

9.17928 

0.15110 

9.19703 

0.15741 

9.21436 

0.16382 

4  46 

.16140 

.14501 

.17958 

.15121 

.19732 

.15751 

.21464 

.16392 

5  47 

.16170 

.14511 

.17988 

.15131 

.19761 

.15762 

.21493 

.16403 

12     48 

.16201 

.14521 

.18018 

.15142 

.19790 

.15773 

.21521 

.16414 

i<5  49 

.16232 

.14532 

.18048 

.15152 

.19820 

.15783 

.21550 

.16425 

;gO  50 

9.16262 

0.14542 

9.18077 

0.15163 

9.19849 

0.15794 

9.21578 

0.16436 

24     51 

.16293 

.14552 

.18107 

.15173 

.19878 

.15804 

.21607 

.16446 

^5  52 

.16324 

.14562 

.18137 

.15183 

.19907 

.15815 

.21635 

.16457 

32     53 

.16354 

.14573 

.18167 

.15194 

.19936 

.15826 

.21664 

.16468 

56  54 

.16385 

.14583 

.18197 

.15204 

.19965 

.15836 

.21692 

.16479 

40  55 

9.16415 

0.14593 

9.18227 

0.15215 

9.19995 

0.15847 

9.21721 

0.16489 

44    56 

.16446 

.14604 

.18256 

.15225 

.20024 

.15858 

.21749 

.16500 

45  57 

.16476 

.14614 

.18286 

.15236 

.20053 

.15868 

.21778 

.16511 

52     58 

.16507 

.14624 

.18316 

.15246 

.20082 

.15879 

.21806 

.16522 

56  59 

.16537 

.14634 

.18346 

.15257 

.20111 

.15889 

.21834 

.16533 

60  60 

9.16568 

0.14645 

9.18376 

0.15267 

9.20140 

0.15900 

9.21863 

0.16543 

262 


Table  10.    Haversine  Table 


s 

1 

gh  12"' 

48° 

Sh  Iffm 

49° 

gh  20"" 

50° 

3h  24"" 

51° 

Hav. 

No. 

Hav. 

No. 

Hav. 

No. 

Hav. 

No. 

0 

0 

9.21863 

0.16543 

9.23545 

0.17197 

9.25190 

0.17861 

9.26797 

0.18534 

4 

1 

.21891 

.16554 

.23573 

.17208 

.25217 

.17872 

.26823 

.18545 

8 

2 

.21919 

.16565 

.23601 

.17219 

.25244 

.17883 

.26850 

.18557 

12 

3 

.21948 

.16576 

.23629 

.17230 

.25271 

.17894 

.26876 

.18568 

16 

4 

.21976 

.16587 

.23656 

.17241 

.25298 

.17905 

.26903 

.18579 

20 

5 

9.22004 

0.16598 

9.23684 

0.17252 

9.25325 

0.17916 

9.26929 

0.18591 

24 

6 

.22033 

.16608 

.23712 

.17263 

.25352 

.17928 

.26956 

.18602 

28 

7 

.22061 

.16619 

.23739 

.17274 

.25379 

.17939 

.26982 

.18613 

32 

8 

.22089 

.16630 

.23767 

.17285 

.25406 

.17950 

.27008 

.18624 

36 

9 

.22118 

.16641 

.23794 

.17296 

.25433 

.17961 

.27035 

.18636 

40 

10 

9.22146 

0.16652 

9.23822 

0.17307 

9.25460 

0.17972 

9.27061 

0.18647 

H 

11 

.22174 

.16663 

.23850 

.17318 

.25487 

.17983 

.27088 

.18658 

48 

12 

.22202 

.16673 

.23877 

.17329 

.25514 

.17995 

.27114 

.18670 

52 

13 

.22231 

.16684 

.23905 

.17340 

.25541 

.18006 

.27140 

.18681 

56 

14 

.22259 

.16695 

.23932 

.17351 

.25568 

.18017 

.27167 

.18692 

s        ' 

Sh  13'" 

48° 

Sh  17m 

49° 

Sh  21'" 

50° 

gh  25'" 

51° 

0 

15 

9.22287 

0.16706 

9.23960 

0.17362 

9.25595 

0.18028 

9.27193 

0.18704 

4 

16 

.22315 

.16717 

.23988 

.17373 

.25622 

.18039 

.27219 

.18715 

8 

17 

.22343 

.16728 

.24015 

.17384 

.25649 

.18050 

.27246 

.18727 

12 

18 

.22372 

.16738 

.24043 

.17395 

.25676 

.18062 

.27272 

.18738 

16 

19 

.22400 

.16749 

.24070 

.17406 

.25703 

.18073 

.27298 

.18749 

20 

20 

9.22428 

0.16760 

9.24098 

0.17417 

9.25729 

0.18084 

9.27325 

0.18761 

24 

21 

.22456 

.16771 

.24125 

.17428 

.25756 

.18095 

.27351 

.18772 

28 

22 

.22484 

.16782 

.24153 

.17439 

.25783 

.18106 

.27377 

.18783 

32 

23 

.22512 

.16793 

.24180 

.17450 

.25810 

.18118 

.27403 

.18795 

36 

24 

.22540 

.16804 

.24208 

.17461 

.25837 

.18129 

.27430 

.18806 

40 

25 

9.22569 

0.16815 

9.24235 

0.17472 

9.25864 

0.18140 

9.27456 

0.18817 

44 

26 

.22597 

.16825 

.24263 

.17483 

.25891 

.18151 

.27482 

.18829 

48 

27 

.22625 

.16836 

.24290 

.17494 

.25917 

.18162 

.27508 

.18840 

52 

28 

.22653 

.16847 

.24317 

.17505 

.25944 

.18174 

.27535 

.18852 

56 

29 

.22681 

.16858 

.24345 

.17517 

.25971 

.18185 

.27561 

.18863 

s 

gh  IJ^m 

48° 

gh  18'" 

49° 

Sh  22'"' 

50° 

Sh  26'" 

51° 

0 

30 

9.22709 

0.16869 

9.24372 

0.17528 

9.25998 

0.18196 

9.27587 

0.18874 

4 

31 

.22737 

.16880 

.24400 

.17539 

.26025 

.18207 

.27613 

.18886 

8 

32 

.22765 

.16891 

.24427 

.17550 

.26051 

.18219 

.27639 

.18897 

12 

33 

.22793 

.16902 

.24454 

.17561 

.26078 

.18230 

.27666 

.18908 

16 

34 

.22821 

.16913 

.24482 

.17572 

.26105 

.18241 

.27692 

.18920 

20 

35 

9.22849 

0.16924 

9.24509 

0.17583 

9.26132 

0.18252 

9.27718 

0.18931 

24 

36 

.22877 

.16934 

.24536 

.17594 

.26158 

.18263 

.27744 

.18943 

28 

37 

.22905 

.16945 

.24564 

.17605 

.26185 

.18275 

.27770 

.18954 

32 

38 

.22933 

.16956 

.24591 

.17616 

.26212 

.18286 

.27796 

.18965 

36 

39 

.22961 

.16967 

.24618 

.17627 

.26238 

.18297 

.27822 

.18977 

40 

40 

9.22989 

0.16978 

9.24646 

0.17638 

9.26265 

0.18308 

9.27848 

0.18988 

44 

41 

.23017 

.16989 

.24673 

.17649 

.26292 

.18320 

.27875 

.19000 

48 

42 

.23045 

.17000 

.24700 

.17661 

.26319 

.18331 

.27901 

.19011 

52 

43 

.23073 

.17011 

.24728 

.17672 

.26345 

.18342 

.27927 

.19022 

56 

44 

.23100 

.17022 

.24755 

.17683 

.26372 

.18353 

.27953 

.19034 

s         ' 

3h  15"^ 

48° 

Sh  ig>n 

49° 

Sh  23'" 

50° 

Sh  27'" 

51° 

0 

45 

9.23128 

0.17033 

9.24782 

0.17694 

9.26398 

0.18365 

9.27979 

0.19045 

4 

46 

.23156 

.17044 

.24809 

.17705 

.26425 

.18376 

.28005 

.19057 

8 

47 

.23184 

.17055 

.24837 

.17716 

.26452 

.18387 

.28031 

.19068 

12 

48 

.23212 

.17066 

.24864 

.17727 

.26478 

.18399 

.28057 

.19080 

16 

49 

.23240 

.17076 

.24891 

.17738 

.26505 

.18410 

.28083 

.19091 

20 

50 

9.23268 

0.17087 

9.24918 

0.17749 

9.26532 

0.18421 

9.28109 

0.19102 

24 

51 

.23295 

.17098 

.24945 

.17760 

.26558 

.18432 

.28135 

.19114 

28 

52 

.23323 

.17109 

.24973 

.17772 

.26585 

.18444 

.28161 

.19125 

32 

53 

.23351 

.17120 

.25000 

.17783 

.26611 

.18455 

.28187 

.19137 

36 

54 

.23379 

.17131 

.25027 

.17794 

.26638 

.18466 

.28213 

.19148 

40 

55 

9.23407 

0.17142 

9.25054 

0.17805 

9.26664 

0.18478 

9.28239 

0.19160 

44 

56 

.23434 

.17153 

.25081 

.17816 

.26691 

.18489 

.28265 

.19171 

48 

57 

.23462 

.17164 

.25108 

.17827 

.26717 

.18500 

.28291 

.19183 

52 

58 

.23490 

.17175 

.25135 

.17«38 

.26744 

.18511 

.28317 

.19194 

56 

59 

.23518 

.17186 

.25163 

.17849 

.26770 

.18523 

.28342 

.19205 

60 

60 

9.23545 

0.17197 

9.25190 

0.17861 

9.26797 

0.18534 

9.28368 

0.19217 

Table  10.    Haversine  Table 


263 


s 

/ 

Sh  28"" 

52° 

5*  32^ 

53° 

gh  Sffm 

54° 

3h  40"" 

55° 

Hav. 

No. 

Hav. 

No. 

Hav. 

No. 

Hav. 

No. 

0 

0 

9.28368 

0.19217 

9.29906 

0.19909 

9.31409 

0.20611 

9.32881 

0.21321 

4 

1 

.28394 

.19228 

.29931 

.19921 

.31434 

.20623 

.32905 

.21333 

8 

2 

.28420 

.19240 

.29956 

.19932 

.31459 

.20634 

.32930 

.21345 

12 

3 

.28446 

.19251 

.29981 

.19944 

.31484 

.20646 

.32954 

.21357 

16 

4 

.28472 

.19263 

.30007 

.19956 

.31508 

.20658 

.32978 

.21369 

20 

5 

9.28498 

0.19274 

9.30032 

0.19967 

9.31533 

0.20670 

9.33002 

0.21381 

24 

6 

.28524 

.19286 

.30057 

.19979 

.31558 

.20681 

.33027 

.21393 

28 

7 

.28549 

.19297 

.30083 

.19991 

.31583 

.20693 

.33051 

.21405 

32 

8 

.28575 

.19309 

.30108 

.20002 

.31607 

.20705 

.33075 

.21417 

36 

9 

.28601 

.19320 

.30133 

.20014 

.31632 

.20717 

.33099 

.21429 

40 

10 

9.28627 

0.19332 

9.30158 

0.20026 

9.31657 

0.20729 

9.33123 

0.21440 

44 

11 

.28653 

.19343 

.30184 

.20037 

.31682 

.20740 

.33148 

.21452 

48 

12 

.28679 

.19355 

.30209 

.20049 

.31706 

.20752 

.33172 

.21464 

52 

13 

.28704 

.19366 

.30234 

.20060 

.31731 

.20764 

.33196 

.21476 

56 

14 

.28730 

.19378 

.30259 

.20072 

.31756 

.20776 

.33220 

.21488 

s 

1 

gh  29"^ 

52° 

Qh    33m 

53° 

3h  3J>n 

54° 

3h  ^jm 

55° 

0 

15 

9.28756 

0.19389 

9.30285 

0.20084 

9.31780 

0.20788 

9.33244 

0.21500 

4 

16 

.28782 

.19401 

.30310 

.20095 

.31805 

.20799 

.33268 

.21512 

8 

17 

.28807 

.19412 

.30335 

.20107 

.31830 

.20811 

.33292 

.21524 

12 

18 

.28833 

.19424 

.30360 

.20119 

.31854 

.20823 

.33317 

.21536 

16 

19 

.28859 

.19435 

.30385 

.20130 

.31879 

.20835 

.33341 

.21548 

20 

20 

9.28885 

0.19447 

9.30410 

0.20142 

9.31903 

0.20847 

9.33365 

0.21560 

24 

21 

.28910 

.19458 

.30436 

.20154 

.31928 

.20858 

.33389 

.21572 

28 

22 

.28936 

.19470 

.30461 

.20165 

.31953 

.20870 

.33413 

.21584 

32 

23 

.28962 

.19481 

.30486 

.20177 

.31977 

.20882 

.33437 

.21596 

36 

24 

.28987 

.19493 

.30511 

.20189 

.32002 

.20894 

.33461 

.21608 

40 

25 

9.29013 

0.19504 

9.30536 

0.20200 

9.32026 

0.20906 

9.33485 

0.21620 

44 

26 

.29039 

.19516 

.30561 

.20212 

.32051 

.20918 

.33509 

.21632 

48 

27 

.29064 

.19527 

.30586 

.20224 

.32076 

.20929 

.33533 

.21644 

52 

28 

.29090 

.19539 

.30011 

.20235 

.32100 

.20941 

.33557 

.21656 

56 

29 

.29116 

.19550 

.30636 

.20247 

.32125 

.20953 

.33581 

.21668 

s 

/ 

Qh  QQm 

52° 

3h  34'" 

53° 

3h  38'n 

54° 

3^^  42'" 

55° 

0 

30 

9.29141 

0.19562 

9.30662 

0.20259 

9.32149 

0.20965 

9.33605 

0.21680 

4 

31 

.29167 

.19573 

.30687 

.20271 

.32174 

.20977 

.33629 

.21692 

8 

32 

.29192 

.19585 

.30712 

.20282 

.32198 

.20989 

.33653 

.21704 

12 

33 

.29218 

.19597 

.30737 

.20294 

.32223 

.21000 

.33677 

.21716 

16 

34 

.29244 

.19608 

.30762 

.20306 

.32247 

.21012 

.33701 

.21728 

20 

35 

9.29269 

0.19620 

9.30787 

0.20317 

9.32272 

0.21024 

9.33725 

0.21740 

24 

36 

.29295 

.19631 

.30812 

.20329 

.32296 

.21036 

.33749 

.21752 

28 

37 

.29320 

.19643 

.30837 

.20341 

.32321 

.21048 

.33773 

.21764 

32 

38 

.29346 

.19654 

.30862 

.20352 

.32345 

.21060 

.33797 

.21776 

36 

39 

.29371 

.19666 

.30887 

.20364 

.32370 

.21072 

.33821 

.21788 

40 

40 

9.29397 

0.19677 

9.30912 

0.20376 

9.32394 

0.21083 

9.33845 

0.21800 

44 

41 

.29422 

.19689 

.30937 

.20388 

.32418 

.21095 

.33869 

.21812 

48 

42 

.29448 

.19701 

.30962 

.20399 

.32443 

.21107 

.33893 

.21824 

52 

43 

.29473 

.19712 

.30987 

.20411 

.32467 

.21119 

.33917 

.21836 

56 

44 

.29499 

.19724 

.31012 

.20423 

.32492 

.21131 

..33941 

.21848 

s 

gh  Spn 

52° 

Sf^  35"" 

53° 

3h  srjm 

9.32516 

54" 
0.21143 

3'^  43'" 

55° 

0 

45 

9.29524 

0.19735 

9.31036 

0.20435 

9.33965 

0.21860 

4 

46 

.29550 

.19747 

.31061 

.20446 

.32541 

.21155 

.33988 

.21872 

8 

47 

.29575 

.19758 

.31086 

.20458 

.32565 

.21167 

.34012 

.21884 

12 

48 

.29601 

.19770 

.31111 

.20470 

.32589 

.21178 

.34036 

.21896 

16 

49 

.29626 

.19782 

.31136 

.20481 

.32614 

.21190 

.34060 

.21908 

20 

50 

9.29652 

0.19793 

9.31161 

0.20493 

9.32638 

0.21202 

9.34084 

0.21920 

24 

51 

.29677 

.19805 

.31186 

.20505 

.32662 

.21214 

.34108 

.21932 

28 

52 

.29703 

.19816 

.31211 

.20517 

.32687 

.21226 

.34132 

.21944 

32 

53 

.29728 

.19828 

.31236 

.20528 

.32711 

.21238 

.34155 

.21956 

36 

54 

.29753 

.19840 

.31260 

.20540 

.32735 

.21250 

.34179 

.21968 

40 

55 

9.29779 

0.19851 

9.31285 

0.20552 

9.32760 

0.21262 

9.34203 

0.21980 

44 

56 

.29804 

.19863 

.31310 

.20564 

.32784 

.21274 

.34227 

.21992 

48 

57 

.29829 

.19874 

.31335 

.20575 

.32808 

.21285 

.34251 

.22004 

52 

58 

.29855 

.19886 

.31360 

.20587 

.32833 

.21297 

.34274 

.22016 

56 

59 

.29880 

.19898 

.31385 

.20599 

.32857 

.21309 

.34298 

.22028 

60 

60 

9.29906 

0.19909 

9.31409 

0.20611 

9.32881 

0.21321 

9.34322 

0.22040 

264 


Table  10.    Haversine  Table 


/ 

3h  44^ 
Hav. 

9.34322 

56° 
No. 

3^  48"" 

57° 

gh   5^m 

58° 

gh  SQrn 

59° 

Hav. 

9.35733 

No. 

Hav. 

9.37114 

No. 

Hav. 

No. 

0 

0 

0.22040 

0.22768 

0.23504 

9.38468 

0.24248 

4 

1 

.34346 

.22052 

.35756 

.22780 

.37137 

.23516 

.38490 

.24261 

8 

2 

.34369 

.22064 

.35779 

.22792 

.37160 

.23529 

.38512 

.24273 

12 

3 

.34393 

.22077 

.35802 

.22805 

.37183 

.23541 

.38535 

.24286 

16 

4 

.34417 

.22089 

.35826 

.22817 

.37205 

.23553 

.38557 

.24298 

20 

5 

9.34441 

0.22101 

9.35849 

0.22829 

9.37228 

0.23566 

9.38579 

0.24310 

24 

6 

.34464 

.22113 

.35872 

.22841 

.37251 

.23578 

.38602 

.24323 

28 

7 

.34488 

.22125 

.35895 

.22853 

.37274 

.23590 

.38624 

.24335 

32 

8 

.34512 

.22137 

.35918 

.22866 

.37296 

.23603 

.38646 

.24348 

36 

9 

.34535 

.22149 

.35942 

.22878 

.37319 

.23615 

.38668 

.24360 

40 

10 

9.34559 

0.22161 

9.35965 

0.22890 

9.37342 

0.23627 

9.38691 

0.24373 

44 

11 

.34583 

.22173 

.35988 

.22902 

.37364 

.23640 

.38713 

.24385 

48 

12 

.34606 

.22185 

.36011 

.22915 

.37387 

.23652 

.38735 

.24398 

52 

13 

.34630 

.22197 

.36034 

.22927 

.37410 

.23665 

.38757 

.24410 

56 

14 

.34654 

.22209 

.36058 

.22939 

.37433 

.23677 

.38780 

.24423 

s        ' 

3h  45'" 
9.34677 

56° 
0.22221 

3h  4.9'" 

57° 

3h  58"" 

58° 

gh  5jm 

59° 

0 

15 

9.36081 

0.22951 

9.37455 

0.23689 

9.38802 

0.24435 

4 

16 

.34701 

.22234 

.36104 

.22964 

.37478 

.23702 

.38824 

.24448 

8 

17 

.34725 

.22246 

.36127 

.22976 

.37501 

.23714 

.38846 

.24460 

12 

18 

,34748 

.22258 

.36150 

.22988 

.37523 

.23726 

.38868 

.24473 

16 

19 

.34772 

.22270 

.36173 

.23000 

.37546 

.23739 

.38891 

.24485 

20 

20 

9.34795 

0.22282 

9.36196 

0.23012 

9.37569 

0.23751 

9.38913 

0.24498 

24 

21 

.34819 

.22294 

.36219 

.23025 

.37591 

.23764 

.38935 

.24510 

28 

22 

.34843 

.22306 

.36243 

.23037 

.37614 

.23776 

.38957 

.24523 

32 

23 

.34866 

.22318 

.36266 

.23049 

.37636 

.23788 

.38979 

.24535 

36 

24 

.34890 

.22330 

.36289 

.23061 

.37659 

.23801 

.39002 

.24548  , 

40 

25 

9.34913 

0.22343 

9.36312 

0.23074 

9.37682 

0.23813 

9.39024 

0.24560 

44 

26 

.34937 

.22355 

.36335 

.23086 

.37704 

.23825 

.39046 

.24573 

48 

27 

.34960 

.22367 

.36358 

.23098 

.37727 

.23838 

.39068 

.24586 

52 

23 

.34984 

.22379 

.36381 

.23110 

.37749 

.23850 

.39090 

.24598 

56 

s 

29 

.35007 

.22391 

.36404 

.23123 

.37772 

.23863 

.39112 

.24611 

3h  46"^ 
9.35031 

56° 
0.22403 

Sh  50"' 
9.36427 

57° 
0.23135 

Sf'  54"" 
9.37794 

58° 
0.23875 

3^  58"' 

59° 

0 

30 

9.39134 

0.24623 

4 

31 

.35054 

.22415 

.36450 

.23147 

.37817 

.23887 

.39156 

.24636 

8 

32 

.35078 

.22427 

.36473 

.23160 

.37840 

.23900 

.39178 

.24648 

12 

33 

.35101 

.22440 

.36496 

.23172 

.37862 

.23912 

.39201 

.24661 

16 

34 

.35125 

.22452 

.36519 

.23184 

.37885 

.23925 

.39223 

.24673 

20 

35 

9.35148 

0.22464 

9.36542 

0.23196 

9.37907 

0.23937 

9.39245 

0.24686 

24 

36 

.35172 

.22476 

.36565 

.23209 

.37930 

.23950 

.39267 

.24698 

28 

37 

.35195 

.22488 

.36588 

.23221 

.37952 

.23962 

.39289 

.24711 

32 

38 

.35219 

.22500 

.36611 

.23233 

.37975 

.23974 

.39311 

.24723 

36 

39 

.35242 

.22512 

.36634 

.23246 

.37997 

.23987 

.39333 

.24736 

40 

40 

9.35266 

0.22525 

9.36657 

0.23258 

9.38020 

0.23999 

9.39355 

0.24749 

44 

41 

.35289 

.22537 

.36680 

.23270 

.38042 

.24012 

.39377 

.24761 

48 

42 

.35312 

.22549 

.36703 

.23282 

.38065 

.24024 

.39399 

.24774 

52 

43 

.35336 

.22561 

.36726 

.23295 

.38087 

.24036 

.39421 

.24786 

56 

44 

.35359 

.22573 
56^ 

.36749 

3h  ol"' 

9.36772 

.23307 

57° 

0.23319 

.38110 

3^^  55"" 
9.38132 

.24049 
58° 
0.24061 

.39443 

.24799 

s         ' 

3''  oO'" 

59° 

0 

45 

9.35383 

0.22585 

9.39465 

0.24811 

4 

46 

.35406 

.22598 

.36794 

.23332 

.38154 

.24074 

.39487 

.24824 

8 

47 

.35429 

.22610 

.36817 

.23344 

.38177 

.24086 

.39509 

.24836 

12 

48 

.35453 

.22622 

.36840 

.23356 

.38199 

.24099 

.39531 

.24849 

16 

49 

.35476 

.22634 

.36863 

.23368 

.38222 

.24111 

.39553 

.24862 

20 

50 

9.35500 

0.22646 

9.36886 

0.23381 

9.38244 

0.24124 

9.39575 

0.24874 

24 

51 

.35523 

.22658 

.36909 

.23393 

.38267 

.24136 

.39597 

.24887 

28 

52 

.35546 

.22671 

.36932 

.23405 

.38289 

.24148 

.39619 

.24899 

32 

53 

.35570 

.22683 

.36955 

.23418 

.38311 

.24161 

.39641 

.24912 

36 

54 

.35593 

.22695 

.36977 

.23430 

.38334 

.24173 

.39663 

.24924 

40 

55 

9.35616 

0.22707 

9.37000 

0.23442 

9.38356 

0.24186 

9.39685 

0.24937 

44 

56 

.35639 

.22719 

.37023 

.23455 

.38378 

.24198 

.39706 

.24950 

48 

57 

.35663 

.22731 

.37046 

.23467 

.38401 

.24211 

.39728 

.24962 

52 

58 

.35686 

.22744 

.37069 

.23479 

.38423 

.24223 

.39750 

.24975 

56 

59 

.35709 

.22756 

.37091 

.23492 

.38445 

.24236 

.39772 

.24987 

60 

60 

9.35733 

0.22768 

9.37114 

0.235C4 

9.38468 

0.24248 

9.39794 

0.25000 

Table  10.    Haversine  Table 


265 


s 

^/tO"* 

60° 

4^^  4"' 

61° 

4h  8"' 

62° 

4h  12"' 

63° 

Hav. 

No. 

Hav. 

No. 

Hav. 

No. 

Hav. 

No. 

0       0 

9.39794 

0.25000 

9.41094 

0.25760 

9.42308 

0.26526 

9.43617 

0.27300 

4      1 

.39816 

.25013 

.41115 

.25772 

.42389 

.26539 

.43638 

.27313 

8       2 

.39838 

.25025 

.41137 

.25785 

.42410 

.26552 

.43658 

.27326 

12       3 

.39860 

.25038 

.41158 

.25798 

.42431 

.26565 

.43679 

.27339 

16       4 

.39881 

.25050 

.41180 

.25810 

.42452 

.26578 

.43699 

.27352 

20       5 

9.39903 

0.25063 

9.41201 

0.25823 

9.42473 

0.26591 

9.43720 

0.27365 

24       6 

.39925 

.25076 

.41222 

.25836 

.42494 

.26604 

.43741 

.27378 

28       7 

.39947 

.25088 

.41244 

.25849 

.42515 

.26616 

.43761 

.27391 

5^   8 

.39969 

.25101 

.41265 

.25861 

.42536 

.26629 

.43782 

.27404 

3tf   9 

.39991 

.25113 

.41287 

.25874 

.42557 

.26642 

.43802 

.27417 

40     10 

9.40012 

0.25126 

9.41308 

0.25887 

9.42578 

0.26655 

9.43823 

0.27430 

44  11 

.40034 

.25139 

.41329 

.25900 

.42599 

.26668 

.43843 

.27443 

48     12 

.40056 

.25151 

.41351 

.25912 

.42620 

.26681 

.43864 

.27456 

5^  13 

.40078 

.25164 

.41372 

.25925 

.42641 

.26694 

.43884 

.27469 

5^  14 

.40100 

.25177 

.41393 

.25938 
61° 

.42662 

.26706 

.43905 

.27482 

s   ' 

4h  i'« 

9.40121 

60° 
0.25189 

9.42682 

62° 
0.26719 

4h   13"^ 

63° 

0     15 

9.41415 

0.25951 

9.43926 

0.27495 

4     16 

.40143 

.25202 

.41430 

.25963 

.42703 

.26732 

.43946 

.27508 

5  17 

.40165 

.25214 

.41457 

.25976 

.42724 

.26745 

.43967 

.27521 

i^  18 

.40187 

.25227 

.41479 

.25989 

.42745 

.26758 

.43987 

.27534 

16     19 

.40208 

.25240 

.41500 

.26002 

.42766 

.26771 

.44008 

.27547 

;gO  20 

9.40230 

0.25252 

9.41521 

0.26014 

9.42787 

0.26784 

9.44028 

0.27560 

24     21 

.40252 

.25265 

.41543 

.26027 

.42808 

.26797 

.44048 

.27573 

;g5  22 

.40274 

.25278 

.41564 

.26040 

.42829 

.26809 

.44069 

.27586 

5^  23 

.40295 

.25290 

.41585 

.26053 

.42850 

.26822 

.44089 

.27599 

36     24 

.40317 

.25303 

.41606 

.26065 

.42870 

.26835 

.44110 

.27612 

40  25 

9.40339 

0.25316 

9.41628 

0.26078 

9.42891 

0.26848 

9.44130 

0.27625 

44     26 

.40360 

.25328 

.41649 

.26091 

.42912 

.26861 

.44151 

.27638 

4<5  27 

.40382 

.25341 

.41670 

.26104 

.42933 

.26874 

.44171 

.27651 

5^  28 

.40404 

.25354 

.41692 

.26117 

.42954 

.26887 

.44192 

.27664 

56     29 

.40425 

.25366 

.41713 

.26129 

.42975 

.26900 

.44212 

.27677 

s   ' 

j^h  2"" 

60° 

^h  Qm 

61° 

4h  10'^ 

62° 

4^  14"' 

63° 

0     30 

9.40447 

0.25379 

9.41734 

0.26142 

9.42996 

0.26913 

9.44232 

0.27690 

4  31 

.40469 

.25391 

.41755 

.26155 

.43016 

.26925 

.44253 

.27703 

5  32 

.40490 

.25404 

.41776 

.26168 

.43037 

.26938 

.44273 

.27716 

12     33 

.40512 

.25417 

.41798 

.26180 

.43058 

.26951 

.44294 

.27729 

i6'  34 

.40534 

.25429 

.41819 

.26193 

.43079 

.26964 

.44314 

.27742 

20     35 

9.40555 

0.25442 

9.41840 

0.26206 

9.43100 

0.26977 

9.44334 

0.27755 

^4  36 

.40577 

.25455 

.41861 

.26219 

.43120 

.26990 

.44355 

.27768 

^<§  37 

.40599 

.25467 

.41882 

.26232 

.43141 

.27003 

.44375 

.27781 

32     38 

.40620 

.25480 

.41904 

.26244 

.43162 

.27016 

.44396 

.27794 

3^  39 

.40642 

.25493 

.41925 

.26257 

.43183 

.27029 

.44416 

.27807 

40     40 

9.40663 

0.25506 

9.41946 

0.26270 

9.43203 

0.27042 

9.44436 

0.27820 

44  41 

.40685 

.25518 

.41967 

.26283 

.43224 

.27055 

A4.4.bl 

.27833 

4<¥  42 

.40707 

.25531 

.41988 

.26296 

.43245 

.27068 

.44:477 

.27846 

52     43 

.40728 

.25544 

.42009 

.26308 

.43266 

.27080 

.44497 

.27859 

5^  44 

.40750 

.25556 

.42031 

4^  7"^ 

.26321 
61° 

.43286 

.27093 

.44518 

.27873 

s 

J^h  S'n 

60° 

4''  11'" 
9.43307 

62° 
0.27106 

4h  15"' 

9.44538 

63° 
0.27886 

0     45 

9.40771 

0.25569 

9.42052 

0.26334 

4  46 

.40793 

.25582 

.42073 

.26347 

.43328 

.27119 

.44558 

.27899 

8     47 

.40814 

.25594 

.42094 

.26360 

.43348 

.27132 

.44579 

.27912 

i:g  48 

.40836 

.25607 

.42115 

.26372 

.43369 

.27145 

.44599 

.27925 

ie  49 

.40858 

.25620 

.42136 

.26385 

.43390 

.27158 

.44619 

.27938 

20     50 

9.40879 

0.25632 

9.42157 

0.26398 

9.43411 

0.27171 

9.44639 

0.27951 

:g4  51 

.40900 

.25645 

.42178 

.26411 

.43431 

.27184 

.44660 

.27964 

28     52 

.40922 

.25658 

.42199 

.26424 

.43452 

.27197 

.44680 

.27977 

5^  53 

.40943 

.25671 

.42221 

.26437 

.43473 

.27210 

.44700 

.27990 

56>  54 

.40965 

.25683 

.42242 

.26449 

.43493 

.27223 

.44721 

.28003 

40     55 

9.40986 

0.25696 

9.42263 

0.26462 

9.4.3514 

0.27236 

9.44741 

0.28016 

44  56 

.41008 

.25709 

.42284 

.26475 

.43535 

.27249 

.44761 

.28029 

48     57 

.41029 

.25721 

.42305 

.26488 

.43555 

.27262 

.44781 

.28042 

5:^  58 

.41051 

.25734 

.42326 

.26501 

.43576 

.27275 

.44801 

.28055 

55  59 

.41072 

.25747 

.42347 

.26514 

.43596 

.27288 

.44822 

.28068 

60     60 

9.41094 

0.25760 

9.42368 

0.26526 

9.43617 

0.27300 

9.44842 

0.28081 

266 


Table  10.    Haversine  Table 


s 

/ 

4h  16"" 

64° 

4''  20'" 
Hav. 

65° 
No. 

4h  24'" 
Hav. 

66° 
No. 

4h  28"^ 

67° 

Hav. 

No. 

Hav. 

No. 

0 

0 

9.44842 

0.28081 

9.46043 

0.28869 

9.47222 

0.29663 

9.48378 

0.30463 

A 

1 

.44862 

.28095 

.46063 

.28882 

.47241 

.29676 

.48397 

.30477 

8 

2 

.44882 

.28108 

.46083 

.28895 

.47261 

.29690 

.48416 

.30490 

12 

3 

.44903 

.28121 

.46103 

.28909 

.47280 

.29703 

.48435 

.30504 

16 

4 

.44923 

.28134 

.46123 

.28922 

.47300 

.29716 

.48454 

.30517 

20 

5 

9.44943 

0.28147 

9.46142 

0.28935 

9.47319 

0.29730 

9.48473 

0.30530 

24 

6 

.44963 

.28160 

.46162 

.28948 

.47338 

.29743 

.48492 

.30544 

28 

7 

.44983 

.28173 

.46182 

.28961 

.47358 

.29756 

.48511 

.30557 

32 

8 

.45003 

.28186 

.46202 

.28975 

.47377 

.29770 

.48530 

.30571 

36 

9 

.45024 

.28199 

.46222 

.28988 

.47397 

.29783 

.48549 

.30584 

40 

10 

9.45044 

0.28212 

9.46241 

0.29001 

9.47416 

0.29796 

9.48568 

0.30597 

U 

11 

.45064 

.28225 

.46261 

.29014 

.47435 

.29809 

.48587 

.30611 

48 

12 

.45084 

.28238 

.46281 

.29027 

.47455 

.29823 

.48607 

.30624 

52 

13 

.45104 

.28252 

.46301 

.29041 

Al^l\ 

.29836 

.48626 

.30638 

56 

14 

.45124 

.28265 

.46320 

.29054 

.47493 

.29849 

.48645 

.30651 

s 

4h  17'" 

64° 

^h  21"' 

65° 

4^  25'" 

66^ 

4h  29'^ 

67° 

0 

15 

9.45144 

0.28278 

9.46340 

0.29067 

9.47513 

0.29863 

9.48664 

0.30664 

4 

16 

.45165 

.28291 

.46360 

.29080 

.47532 

.29876 

.48683 

.30678 

8 

17 

.45185 

.28304 

.46380 

.29093 

.47552 

.29889 

.48702 

.30691 

12 

18 

.45205 

.28317 

.46399 

.29107 

.47571 

.29903 

.48720 

.30705 

16 

19 

.45225 

.28330 

.46419 

.29120 

.47590 

.29916 

.48739 

.30718 

20 

20 

9.45245 

0.28343 

9.46439 

0.29133 

9.47610 

0.29929 

9.48758 

0.30732 

24 

21 

.45265 

.28356 

.46458 

.29146 

.47629 

.29943 

.48777 

.30745 

28 

22 

.45285 

.28369 

.46478 

.29160 

.47648 

.29956 

.48796 

.30758 

32 

23 

.45305 

.28383 

.46498 

.29173 

.47668 

.29969 

.48815 

.30772 

36 

24 

.45325 

.28396 

.46517 

.29186 

.47687 

.29983 

.48834 

.30785 

40 

25 

9.45345 

0.28409 

9.40537 

0.29199 

9.47706 

0.29996 

9.48853 

0.30799 

44 

26 

.45365 

.28422 

.46557 

.29212 

.47725 

.30009 

.48872 

.30812 

48 

27 

.45385 

.28435 

.46576 

.29226 

.47745 

.30023 

.48891 

.30826 

52 

28 

.45405 

.28448 

.46596 

.29239 

AllQA 

.30036 

.48910 

.30839 

56 

29 

.45426 

.28461 

.46616 

.29252 

.47783 

.30049 

.48929 

.30852 

s 

4/^  18'" 

64° 

4h  22'" 

65° 

4h  26'" 

66° 

4h  30'" 

67° 

0 

30 

9.45446 

0.28474 

9.46635 

0.29265 

9.47803 

0.30063 

9.48948 

0.30866 

4 

31 

.45466 

.28488 

.46655 

.29279 

.47822 

.30076 

.48967 

.30879 

8 

32 

.45486 

.28501 

.46675 

.29292 

.47841 

.30089 

.48986 

.30893 

12 

33 

.45506 

.28514 

.46694 

.29305 

.47860 

.30103 

.49004 

.30906 

16 

34 

.45526 

.28527 

.46714 

.29318 

.47880 

.30116 

.49023 

.30920 

20 

35 

9.45546 

0.28540 

9.46733 

0.29332 

9.47899 

0.30129 

9.49042 

0.30933 

24 

36 

.45566 

.28553 

.46753 

.29345 

.47918 

.30143 

.49061 

.30946 

28 

37 

.45586 

.28566 

.46773 

.29358 

.47937 

.30156 

•49080 

.30960 

32 

38 

.45606 

.28580 

.46792 

.29371 

.47957 

.30169 

.49099 

.30973 

36 

39 

.45625 

.28593 

.46812 

.29385 

.47976 

.30183 

.49118 

.30987 

40 

40 

9.45645 

0.28606 

9.46831 

0.29398 

9.47995 

0.30196 

9.49137 

0.31000 

44 

41 

.45665 

.28619 

.46851 

.29411 

.48014 

.302Q9 

.49155 

.31014 

48 

42 

.45685 

.28632 

.46871 

.29424 

.48033 

.30223 

.49174 

.31027 

52 

43 

.45705 

.28645 

.46890 

.29438 

.48053 

.30236 

.49193 

.31041 

56 

44 

.45725 

.28658 

.46910 

.29451 

.48072 

.30249 

.49212 

.31054 

s        ' 

4h  i,9"» 
9.45745 

64° 
0.28672 

4h  23'" 

65° 

4^  27'" 

66° 

4h  31'" 

67° 

0 

45 

9.46929 

0.29464 

9.48091 

0.30263 

9.49231 

0.31068 

4 

46 

.45765 

.28685 

.46949 

.29477 

.48110 

.30276 

.49250 

.31081 

8 

47 

.45785 

.28698 

.46968 

.29491 

.48129 

.30290 

.49268 

.31095 

12 

48 

.45805 

.28711 

.46988 

.29504 

.48148 

.30303 

.49287 

.31108 

16 

49 

.45825 

.28724 

.47007 

.29517 

.48168 

.30316 

.49306 

.31121 

20 

50 

9.45845 

0.28737 

9.47027 

0.29530 

9.48187 

0.30330 

9.49325 

0.31135 

24 

51 

.45865 

.28751 

.47046 

.29544 

.48206 

.30343 

.49344 

.31148 

28 

52 

.45884 

.28764 

.47066 

.29557 

.48225 

.30356 

.49362 

.31162 

32 

53 

.45904 

.28777 

.47085 

.29570 

.48244 

.30370 

.49481 

.31175 

36 

54 

.45924 

.28790 

.47105 

.29583 

.48263 

.30383 

.49400 

.31189 

40 

55 

9.45944 

0.28803 

9.47124 

0.29597 

9.48282 

0.30397 

9.49419 

0.31202 

U 

56 

.45964 

.28816 

.47144 

.29610 

.48302 

.30410 

.49437 

.31216 

48 

57 

.45984 

.28830 

.47163 

.29623 

.48321 

.30423 

.49456 

.31229 

52 

58 

.46004 

.28843 

,47183 

.29637 

.48340 

.30437 

.49475 

.31243 

56 

59 

.46023 

.28856 

.47202 

.29650 

.48359 

.30450 

.49494 

.31256 

60 

60 

9.46043 

0.28869 

9.47222 

0.29663 

9.48378 

0.30463 

9.49512 

0.31270 

Table  10.    Hayersine  Table 


261 


s 

4h  32"" 

68° 

4h  36^ 

69° 

4h  40'" 

70° 

4^44"" 
Hav. 

71° 
No. 

Hav. 

No. 

Hav. 

No. 

Hav. 

No. 

0       0 

9.49512 

0.31270 

9.50626 

0.32082 

9.51718 

0.32899 

9.52791 

0.33722 

4   1 

.49531 

.31283 

.50644 

.32095 

.51736 

.32913 

.52809 

.33735 

8       2 

.49550 

.31297 

.50662 

.32109 

.51754 

.32926 

.52826 

.33749 

12       3 

.49568 

.31310 

.50681 

.32122 

.51772 

.32940 

.52844 

.33763 

16       4 

.49587 

.31324 

.50699 

.32136 

.51790 

.32954 

.52862 

.33777 

20       5 

9.49606 

0.31337 

9.50717 

0.32150 

9.51808 

0.32967 

9.52879 

0.33790 

24       6 

.49625 

.31351 

.50736 

.32163 

.51826 

.32981 

.52897 

.33804 

28       7 

.49643 

.31364 

.50754 

.32177 

.51844 

.32995 

.52915 

.33818 

5^   8 

.49662 

.31378 

.50772 

.32190 

.51862 

.33008 

.52932 

.33832 

5^   9 

.49681 

.31391 

.50791 

.32204 

.51880 

.33022 

.52950 

.33845 

40     10 

9.49699 

0.31405 

9.50809 

0.32217 

9.51898 

0.33036 

9.52968 

0.33859 

44  11 

.49718 

.31418 

.50827 

.32231 

.51916 

.33049 

.52985 

.33873 

48    12 

.49737 

.31432 

.50846 

.32245 

.51934 

.33063 

.53003 

.33887 

J^  13 

.49755 

.31445 

.50864 

.32258 

.51952 

.33077 

.53021 

.33900 

5^  14 

.49774 

.31459 

.50882 

.32272 

.51970 

.33090 

.53038 

.33914 

s 

4h  33'" 

68" 

4h  37'" 

69° 

4h  4V" 

70° 

4'*  45'" 
9.53056 

71° 
0.33928 

0     15 

9.49793 

0.31472 

9.50901 

0.32285 

9.51988 

0.33104 

4  16 

.49811 

.31486 

.50919 

.32299 

.52006 

.33118 

.53073 

.33942 

8     17 

.49830 

.31499 

.50937 

.32313 

.52024 

.33132 

.53091 

.33956 

i^  18 

.49849 

.31513 

.50956 

.32326 

.52042 

.33145 

.53109 

.33969 

16     19 

.49867 

.31526 

.50974 

.32340 

.52060 

.33159 

.53126 

.33983 

^0  20 

9.49886 

0.31540 

9.50992 

0.32353 

9.52078 

0.33173 

9.53144 

0.33997 

^4  21 

.49904 

.31553 

.51010 

.32367 

.52096 

.33186 

.53162 

.34011 

^5  22 

.49923 

.31567 

.51029 

.32381 

.52114 

.33200 

.53179 

.34024 

5^  23 

.49942 

.31580 

.51047 

.32394 

.52132 

.33214 

.53197 

.34038 

36     24 

.49960 

.31594 

.51065 

.32408 

.52150 

.33227 

.53214 

.34052 

40  25 

9.49979 

0.31607 

9.51083 

0.32422 

9.52168 

0.33241 

9.53232 

0.34066 

44  26 

.49997 

.31621 

.51102 

.32435 

.52185 

.33255 

.53249 

.34080 

48     27 

.50016 

.31634 

.51120 

.32449 

.52203 

.33269 

.53267 

.34093 

o;g  28 

.50034 

.31648 

.51138 

.32462 

.52221 

.33282 

.53285 

.34107 

56     29 

.50053 

.31661 

.51156 

.32476 

.52239 

.33296 

.53302 

.34121 

s 

4h  34'" 

68° 

4^38'" 

69° 

4^  42'" 

70° 

4h  46"' 

71° 

0  30 

9.50072 

0.31675 

9.51174 

0.32490 

9.52257 

0.33310 

9.53320 

0.34135 

4  31 

.50090 

.31688 

.51193 

.32503 

.52275 

.33323 

.53337 

.34149 

5  32 

.50109 

.31702 

.51211 

.32517 

.52293 

.33337 

.53355 

.34162 

i^  33 

.50127 

.31716 

.51229 

.32531 

.52311 

.33351 

.53372 

.34176 

16     34 

.50146 

.31729 

.51247 

.32544 

.52328 

.33365 

.53390 

.34190 

:eO  35 

9.50164 

0.31742 

9.51265 

0.32558 

9.52346 

0.33378 

9.53407 

0.34204 

24     36 

.50183 

.31756 

.51284 

.32571 

.52364 

.33392 

.53425 

.34218 

^5  37 

.50201 

.31770 

.51302 

.32585 

.52382 

.33406 

.53442 

.34231 

5^  38 

.50220 

.31783 

.51320 

.32599 

.52400 

.33419 

.53460 

.34245 

36     39 

.50238 

.31797 

.51338 

.32612 

.52418 

.33433 

.53477 

.34259 

40  40 

9.50257 

0.31810 

9.51356 

0.32626 

9.52436 

0.33447 

9.53495 

0.34273 

44    41 

.50275 

.31824 

.51374 

.32640 

.52453 

.33461 

.53512 

.34287 

4<§  42 

.50294 

.31837 

.51393 

.32653 

.52471 

.33474 

.53530 

.34300 

J^  43 

.50312 

.31851 

.51411 

.32667 

.52489 

.33488 

.53547 

.34314 

56     44 

.50331 

.31865 

.51429 

.32681 

.52507 

.33502 

.53565 

.34328 

s 

4^  35'" 

68° 

4h  39'" 

69° 

4^43"^ 

70° 

4''  47"" 

71° 

0  45 

9.50349 

0.31878 

9.51447 

0.32694 

9.52525 

0.33515 

9.53582 

0.34342 

4  46 

.50368 

.31892 

.51465 

.32708 

.52542 

.33529 

.53600 

.34356 

<5  47 

.50386 

.31905 

.51483 

.32721 

.52560 

.33543 

.53617 

.34369 

12     48 

.50405 

.31919 

.51501 

.32735 

.52578 

.33557 

.53635 

.34383 

i^  49 

.50423 

.31932 

.51519 

.32749 

.52596 

.33570 

.53652 

.34397 

^0  50 

9.50442 

0.31946 

9.51538 

0.32762 

9.52613 

0.33584 

9.53670 

0.34411 

24     51 

.50460 

.31959 

.51556 

.32776 

.52631 

.33598 

.53687 

.34425 

^5  52 

.50478 

.31973 

.51574 

.32790 

.52649 

.33612 

.53704 

.34439 

32     53 

.50497 

.31987 

.51592 

.32803 

.52667 

.33625 

.53722 

.34452 

35  54 

.50515 

.32000 

.51610 

.32817 

.52684 

.33639 

.53739 

.34466 

40  55 

9.50534 

0.32014 

9.51628 

0.32831 

9.52702 

0.33653 

9.53757 

0.34480 

44     56 

.50552 

.32027 

.51646 

.32844 

.52720 

.33667 

.53774 

.34494 

45  57 

.50570 

.32041 

.51664 

.32858 

.52738 

.33680 

.53792 

.34508 

52     58 

.50589 

.32054 

.51682 

.32872 

.52755 

.33694 

.53809 

.34521 

55  59 

.50607 

.32068 

.51700 

.32885 

.52773 

.33708 

.53826 

.34535 

50  60 

9.50626 

0.32082 

9.51718 

0.32899 

9.52791 

0.33722 

9.53844 

0.34549 

268 


Table  10.    Haversine  Table 


s 

1 

4^  48'" 

72° 

4^^  52'" 

73° 

4^  56'" 

74° 

5h  0^ 

75° 

Hav. 

No. 

Hav. 

No. 

Hav. 

No. 

Hav. 

No. 

0 

0 

9.53844 

0.34549 

9.54878 

0.35381 

9.55893 

0.36218 

9.56889 

0.37059 

4 

1 

.53861 

.34563 

.54895 

.35395 

.55909 

.36232 

.56906 

.37073 

8 

2 

.53879 

.34577 

.54912 

.35409 

.55926 

.36246 

.56922 

.37087 

12 

3 

.53896 

.34591 

.54929 

.35423 

.55943 

.36260 

.56939 

.37101 

16 

4 

.53913 

.34604 

.54946 

.35437 

.55960 

.36274 

.56955 

.37115 

20 

5 

9.53931 

0.34618 

9.54963 

0.35451 

9.55976 

0.36288 

9.56972 

0.37129 

24 

6 

.53948 

.34632 

.54980 

.35465 

.55993 

.36302 

.56988 

.37143 

28 

7 

.53966 

.34646 

.54997 

.35479 

.56010 

.36316 

.57005 

.37157 

32 

8 

.53983 

.34660 

.55014 

.35493 

.56027 

.36330 

.57021 

.37171 

36 

9 

.54000 

.34674 

.55031 

.35507 

.56043 

.36344 

.57037 

.37186 

40 

10 

9.54017 

0.34688 

9.55048 

0.35521 

9.56060 

0.36358 

9.57054 

0.37200 

44 

11 

.54035 

.34701 

.55065 

.35534 

.56077 

.36372 

.57070 

.37214 

48 

12 

.54052 

.34715 

.55082 

.35548 

.56093 

.36386 

.57087 

.37228 

52 

13 

.54069 

.34729 

.55099 

.35562 

.56110 

.36400 

.57103 

.37242 

56 

14 

.54087 

.34743 

.55116 

.35576 

.56127 

.36414 

.57119 

.37256 

s 

4h  49'" 

72° 

4h  53'" 

73" 

4/1  5Jm 

74° 

gh  jm 

75° 

0 

15 

9.54104 

0.34757 

9.55133 

0.35590 

9.56144 

0.36428 

9.57136 

0.37270 

4 

16 

.54121 

.34771 

.55150 

.35604 

.56160 

.36442 

.57152 

.37284 

8 

17 

.54139 

.34784 

.55167 

.35618 

.56177 

.36456 

.57169 

.37298 

12 

18 

.54156 

.34798 

.55184 

.35632 

.56194 

.36470 

.57185 

.37312 

16 

19 

.54173 

.34812 

.55201 

.35646 

.56210 

.36484 

.57201 

.37326 

20 

20 

9.54190 

0.34826 

9.55218 

0.35660 

9.56227 

0.36498 

9.57218 

0.37340 

24 

21 

.54208 

.34840 

.55235 

.35674 

.56244 

.36512 

.57234 

.37354 

28 

22 

.54225 

.34854 

.55252 

.35688 

.56260 

.36526 

.57250 

.37368 

32 

23 

.54242 

.34868 

.55269 

.35702 

.56277 

.36540 

.57267 

.37382 

36 

24 

.54260 

.34882 

.55280 

.35716 

.56294 

.36554 

.57283 

.37397 

40 

25 

9.54277 

0.34895 

9.55303 

0.35730 

9.56310 

0.36568 

9.57299 

0.37411 

44 

26 

.54294 

.34909 

.55320 

.35743 

.56327 

.36582 

.57316 

.37425 

48 

27 

.54311 

.34923 

.55337 

.35757 

.56343 

.36596 

.57332 

.37439 

62 

28 

.54329 

.34937 

.55354 

.35771 

.56360 

.36610 

.57348 

.37453 

66 

29 

.54346 

.34951 

.55370 

.35785 

.56377 

.36624 

.57365 

.37467 

s 

4^50"" 

72° 

4''  54'" 

73° 

Ah  58'" 

74° 

gh  £m 

75° 

0 

30 

9.54363 

0.34965 

9.55387 

0.35799 

9.56393 

0.36638 

9.57381 

0.37481 

4 

31 

.54380 

.34979 

.55404 

.35813 

.56410 

.36652 

.57397 

.37495 

8 

32 

.54397 

.34992 

.55421 

.35827 

.56426 

.36666 

.57414 

.37509 

12 

33 

.54415 

.35006 

.55438 

.35841 

.56443 

.36680 

.57430 

.37523 

16 

34 

.54432 

.35020 

.55455 

.35855 

.56460 

.36694 

.57446 

.37537 

20 

35 

9.54449 

0.35034 

9.55472 

0.35869 

9.56476 

0.36708 

9.57463 

0.37551 

24 

36 

.54466 

.35048 

.55489 

.35883 

.56493 

.36722 

.57479 

.37566 

28 

37 

.54483 

.35062 

.55506 

.35897 

.56509 

.36736 

.57495 

.37580 

32 

38 

.54501 

.35076 

.55523 

.35911 

.56526 

.36750 

.57511 

.37594 

36 

39 

.54518 

.35090 

.55539 

.35925 

.56543 

.36764 

.57528 

.37608 

40 

40 

9.54535 

0.35103 

9.55556 

0.35939 

9.56559 

0.36778 

9.57544 

0.37622 

44 

41 

.54552 

.35117 

.55573 

.35953 

.56576 

.36792 

.57560 

.37636 

48 

42 

.54569 

.35131 

.55590 

.35967 

.56592 

.36806 

.57577 

.37650 

62 

43 

.54587 

.35145 

.55607 

.35981 

.56609 

.36820 

.57593 

.37664 

66 

44 

.54604 

.35159 

.55624 

.35995 

.56625 

.36834 

.57609 

.37678 

s 

/ 

4h  51'" 

72° 

4h  55'" 

73° 

4^  59'" 
9.56642 

74° 
0.36848 

gh  s"' 

75° 

0 

45 

9.54621 

0.35173 

9.55641 

0.36009 

9.57625 

0.37692 

4 

46 

.54638 

.35187 

.55657 

.36023 

.56658 

.36862 

.57642 

.37706 

8 

47 

.54655 

.35201 

.55674 

.36036 

.56675 

.36877 

.57658 

.37721 

12 

48 

.54672 

.35215 

.55691 

.36050 

.56692 

.36891 

.57674 

.37735 

16 

49 

.54689 

.35228 

.55708 

.36064 

.56708 

.36905 

.57690 

.37749 

20 

50 

9.54707 

0.35242 

9.55725 

0.36078 

9.56725 

0.36919 

9.57706 

0.37763 

24 

51 

.54724 

.35256 

.55742 

.36092 

.56741 

.36933 

.57723 

.37777 

28 

52 

.54741 

.35270 

.55758 

.36106 

.56758 

.36947 

.57739 

.37791 

32 

53 

.54758 

.35284 

.55775 

.36120 

.56774 

.36961 

.57755 

.37805 

36 

54 

.54775 

.35298 

.55792 

.36134 

.56791 

.36975 

.57771 

.37819 

40 

55 

9.54792 

0.35312 

9.55809 

0.36148 

9.56807 

0.36989 

9.57787 

0.37833 

44 

56 

.54809 

.35326 

.55826 

.36162 

.56824 

.37003 

.57804 

.37847 

48 

57 

.54826 

.35340 

.55842 

.36176 

.56840 

.37017 

.57820 

.37862 

52 

58 

.54843 

.35354 

.55859 

.36190 

.56856 

.37031 

.57836 

.37876 

56 

59 

.54860 

.35368 

.55876 

.36204 

.56873 

.37045 

.57852 

.37890 

60 

60 

9.54878 

0.35381 

9.55893 

0.36218 

9.56889 

0.37059 

9.57868 

0.37904 

Table  10.    Haversine  Table 


269 


s 

5h   4'" 

76° 

5h  8"' 

77° 

gh  12m 

78° 

gh  10m 

79° 

Hav. 

No. 

Hav. 

9.58830 

No. 

Hav. 

No. 

Hav. 

No. 

0 

0 

9.57868 

0.37904 

0.38752 

9.59774 

0.39604 

9.60702 

0.40460 

4 

1 

.57885 

.37918 

.58846 

.38767 

.59790 

.39619 

.60717 

.40474 

8 

2 

.57901 

.37932 

.58862 

.38781 

.59806 

.39633 

.60733 

.40488 

12 

3 

.57917 

.37946 

.58878 

.38795 

.59821 

.39647 

.60748 

.40502 

16 

4 

.57933 

.37960 

.58893 

.38809 

.59837 

.39661 

.60763 

.40517 

20 

5 

9.57949 

0.37974 

9.58909 

0.38823 

9.59852 

0.39676 

9.60779 

0.40531 

24 

6 

.57965 

.37989 

.58925 

.38837 

.59868 

.39690 

.60794 

.40545 

28 

7 

.57981 

.38003 

.58941 

.38852 

.59883 

.39704 

.60809 

.40560 

32 

8 

.57998 

.38017 

.58957 

.38866 

.59899 

.39718 

.60825 

.40574 

36 

9 

.58014 

.38031 

.58973 

.38880 

.59915 

.39732 

.60840 

.40588 

40 

10 

9.58030 

0.38045 

9.58989 

0.38894 

9.59930 

0.39746 

9.60855 

0.40602 

44 

11 

.58046 

.38059 

.59004 

.38908 

.59940 

.39761 

.60870 

.40617 

48 

12 

.58062 

.38073 

.59020 

.38923 

.59961 

.39775 

.60886 

.40631 

52 

13 

.58078 

.38087 

.59036 

.38937 

.59977 

.39789 

.60901 

.40645 

56 

14 

.58094 

.38102 

.59052 

.38951 

.59992 

.39803 

.60916 

.40660 

s        ' 

5^  5'" 

76° 

gh  gm 

77° 

gh  igyn 

78° 

gh  Ijm 

79° 

0 

15 

9.58110 

0.38116 

9.59068 

0.38965 

9.60008 

0.39818 

9.60931 

0.40674 

4 

16 

.58126 

.38130 

.59083 

.38979 

.60023 

.39832 

.60947 

.40688 

8 

17 

.58143 

.38144 

.59099 

.38994 

.60039 

.39846 

.60962 

.40702 

12 

18 

.58159 

.38158 

.59115 

.39008 

.60054 

.39861 

.60977 

.40717 

16 

19 

.58175 

.38172 

.59131 

.39022 

.60070 

.39875 

.60992 

.40731 

20 

20 

9.58191 

0.38186 

9.59147 

0.39036 

9.60085 

0.39889 

9.61008 

0.40745 

24 

21 

.58207 

.38200 

.59162 

.39050 

.60101 

.39903 

.01023 

.40760 

28 

22 

.58223 

.38215 

.59178 

.39064 

.60116 

.39918 

.61038 

.40774 

32 

23 

.58239 

.38229 

.59194 

.39079 

.60132 

.39932 

.61053 

.40788 

36 

24 

.58255 

.38243 

.59210 

.39093 

.60147 

.39946 

.61069 

.40802 

40 

25 

9.58271 

0.38257 

9.59225 

0.39107 

9.00163 

0.39960 

9.61084 

0.40817 

44 

26 

.58287 

.38271 

.59241 

.39121 

.60178 

.39975 

.61099 

.40831 

48 

27 

.58303 

.38285 

.59257 

.39135 

.60194 

.39989 

.61114 

.40845 

52 

28 

.58319 

.38299 

.59273 

.39150 

.60209 

.40003 

.61129 

.40860 

56 

29 

.58335 

.38314 

.59289 

.39164 

.60225 

.40017 

.61145 

.40874 

s 

Sh  6-'" 

76° 

oh  10'" 

77° 

.5''  14'" 

78° 

gh  IS'" 

79° 

0 

30 

9.58351 

0.38328 

9.59304 

0.39178 

9.60240 

0.40032 

9.61160 

0.40888 

4 

31 

.58367 

.38342 

.59320 

.39192 

.60256 

.40046 

.61175 

.40903 

8 

32 

.58383 

.38356 

.59336 

.39206 

.60271 

.40060 

.61190 

.40917 

12 

33 

.58399 

.38370 

.59351 

.39221 

.60287 

.40074 

.61205 

.40931 

16 

34 

.58415 

.38384 

.59367 

.39235 

.60302 

.40089 

.61221 

.40945 

20 

35 

9.58431 

0.38398 

9.59383 

0.39249 

9.60318 

0.40103 

9.61236 

0.40960 

24 

36 

.58447 

.38413 

.59399 

.39263 

.60333 

.40117 

.61251 

.40974 

28 

37 

.58463 

.38427 

.59414 

.39277 

.60348 

.40131 

.61266 

.40988 

32 

38 

.58479 

.38441 

.59430 

.39292 

.60364 

.40146 

.61281 

.41003 

36 

39 

.58495 

.38455 

.59446 

.39306 

.60379 

.40160 

.61296 

.41017 

40 

40 

9.58511 

0.38469 

9.59461 

0.39320 

9.60395 

0.40174 

9.61312 

0.41031 

44 

41 

.58527 

.38483 

.59477 

.39334 

.60410 

.40188 

.61327 

.41046 

48 

42 

.58543 

.38498 

.59493 

.39348 

.60426 

.40203 

.61342 

.41060 

52 

43 

.58559 

.38512 

.59508 

.39363 

.60441 

.40217 

.61357 

.41074 

56 

44 

.58575 

.38526 

.59524 

.39377 

.60456 

.40231 

.61372 

.41089 

s 

gh  7m 

76° 

gh  11m 

77° 

gh  Ijm 

78° 

5h  19'" 

79° 

0 

45 

9.58591 

0.38540 

9.59540 

0.39391 

9.60472 

0.40245 

9.61387 

0.41103 

4 

46 

.58607 

.38554 

.59556 

.39405 

.60487 

.40260 

.61402 

.41117 

8 

47 

.58623 

.38568 

.59571 

.39420 

.60502 

.40274 

.61417 

.41131 

12 

48 

.58639 

.38582 

.59587 

.39434 

.60518 

.40288 

.61433 

.41146 

16 

49 

.58655 

.38597 

.59602 

.39448 

.60533 

.40303 

.61448 

.41160 

20 

50 

9.58671 

0.38611 

9.59618 

0.39462 

9.60549 

0.40317 

9.61463 

0.41174 

24 

51 

.58687 

.38625 

.59634 

.39476 

.60564 

.40331 

.61478 

.41189 

28 

52 

.58703 

.38639 

.59649 

.39491 

.60579 

.40345 

.61493 

.41203 

32 

53 

.58719 

.38653 

.59665 

.39505 

.60595 

.40360 

.61508 

.41217 

36 

54 

.58735 

.38667 

.59681 

.39519 

.60610 

.40374 

.61523 

.41232 

40 

55 

9.58750 

0.38682 

9.59696 

0.39533 

9.60625 

0.40388 

9.61538 

0.41246 

44 

56 

.58766 

.38696 

.59712 

.39548 

.60641 

.40402 

.61553 

.41260 

48 

57 

.58782 

.38710 

.59728 

.39562 

.60656 

.40417 

.61568 

.41275 

52 

58 

.58798 

.38724 

.59743 

.39576 

.60671 

.40431 

.61583 

.41289 

56 

59 

.58814 

.38738 

.59759 

.39590 

.60687 

.40445 

.61598 

.41303 

60 

60 

9.58830 

0.38752 

9.59774 

0.39604 

9.60702 

0.40460 

9.61614 

0.41318  1 

270 


Table  10.    Haversine  Table 


s 

Qh  20"^ 

80° 

gh  24'^' 

81° 

of^  28"' 

82° 

.jh  32"' 

83° 

Hav. 

No. 

Hav. 

No. 

Hav. 

9.63389 

No. 

Hav. 

No. 

0 

0 

9.61614 

0.41318 

9.62509 

0.42178 

0.43041 

9.64253 

0.43907 

4 

1 

.61629 

.41332 

.62524 

.42193 

.63403 

.43056 

.64267 

.43921 

8 

2 

.61644 

.41346 

.62538 

.42207 

.63418 

.43070 

.64281 

.43935 

12 

3 

.61659 

.41361 

.62553 

.42221 

.63432 

.43085 

.64296 

.43950 

16 

4 

.61674 

.41375 

.62568 

.42236 

.63447 

.43099 

.64310 

.43964 

20 

5 

9.61689 

0.41389 

9.62583 

0.42250 

9.63461 

0.43113 

0.64324 

0.43979 

24 

6 

.61704 

.41404 

.62598 

.42264 

.63476 

.43128 

.64339 

.43993 

28 

7 

.61719 

.41418 

.62612 

.42279 

.63490 

.43142 

.64353 

.44008 

32 

8 

.61734 

.41432 

.62627 

.42293 

.63505 

.43157 

.64367 

.44022 

36 

9 

.61749 

.41447 

.62642 

.42308 

.63519 

.43171 

.64381 

.44036 

40 

10 

9.61764 

0.41461 

9.62657 

0.42322 

9.63534 

0.43185 

9.64396 

0.44051 

44 

11 

.61779 

.41475 

.62671 

.42336 

.63548 

.43200 

.64410 

.44065 

48 

12 

.61794 

.41490 

.62686 

.42351 

.63563 

.43214 

.64424 

.44080 

52 

13 

.61809 

.41504 

.62701 

.42365 

.63577 

.43229 

.64438 

.44094 

56 

14 

.61824 

.41518 

.62716 

.42379 

.63592 

gh  29"' 

.43243 
82° 

.64452 
o>^  33"' 

.44109 
83° 

s 

gh  21"^ 

80° 

gh  2o"' 

81° 

0 

15 

9.61839 

0.41533 

9.62730 

0.42394 

9.63606 

0.43257 

9.64467 

0.44123 

4 

16 

.61854 

.41547 

.62745 

.42408 

.63621 

.43272 

.64481 

.44138 

8 

17 

.61869 

.41561 

.62760 

.42423 

.63635 

.43286 

.64495 

.44152 

12 

18 

.61884 

.41576 

.62774 

.42437 

.63649 

.43301 

.64509 

.44166 

16 

19 

.61899 

.41590 

.62789 

.42451 

.63664 

.43315 

.64523 

.44181 

20 

20 

9.61914 

0.41604 

9.62804 

0.42466 

9.63678 

0.43330 

9.64538 

0.44195 

24 

21 

.61929 

.41619 

.62819 

.42480 

.63693 

.43344 

.64552 

.44210 

28 

22 

.61944 

.41633 

.62833 

.42494 

.63707 

.43358 

.64566 

.44224 

32 

23 

.61959 

.41647 

.62848 

.42509 

.63722 

.43373 

.64580 

.44239 

36 

24 

.61974 

.41662 

.62863 

.42523 

.63736 

.43387 

.64594 

.44253 

40 

25 

9.61989 

0.41676 

9.62877 

0.42538 

9.63751 

0.43402 

9.64609 

0.44268 

44 

26 

.62003 

.41690 

.62892 

.42552 

.63765 

.43416 

.64623 

.44282 

48 

27 

.62018 

.41705 

.02907 

.42566 

.63779 

.43430 

.64637 

.44296 

52 

28 

.62033 

.41719 

.62921 

.42581 

.63794 

.43445 

.64651 

.44311 

56 

29 

.62048 

.41733 

.02936 

.42595 

.63808 

.43459 

.64665 

oh  34"' 

.44325 
83° 

s        ' 

gh  22"" 

80° 

gh  2(fn 

81° 

5^  30"" 

82° 

0 

30 

9.62063 

0.41748 

9.62951 

0.42610 

9.63823 

0.43474 

9.64679 

0.44340 

4 

31 

.62078 

.41762 

.62965 

.42624 

.63837 

.43488 

.64694 

.44354 

8 

32 

.62093 

.41776 

.62980 

.42638 

.63851 

.43503 

.64708 

.44369 

12 

33 

.62108 

.41791 

.02995 

.42653 

.63866 

.43517 

.64722 

.44383 

16 

34 

.62123 

.41805 

.63009 

.42667 

.63880 

.43531 

.64736 

.44398 

20 

35 

9.62138 

0.41819 

9.63024 

0.42681 

9.63895 

0.43546 

9.64750 

0.44412 

24 

36 

.62153 

.41834 

.03039 

.42696 

.63909 

.43560 

.64764 

.44427 

28 

37 

.62168 

.41848 

.63063 

.42710 

.63923 

.43575 

.64778 

.44441 

32 

38 

.62182 

.41862 

.63068 

.42725 

.63938 

.43589 

.64793 

.44455 

36 

39 

.62197 

.41877 

.63082 

.42739 

.63952 

.43603 

.64807 

.44470 

40 

40 

9.62212 

0.41891 

9.63097 

0.42753 

9.63966 

0.43618 

9.64821 

0.44484 

44 

41 

.62227 

.41905 

.63112 

.42768 

.63981 

.43632 

.64835 

.44499 

48 

42 

.62242 

.41920 

.63126 

.42782 

.63995 

.43647 

.64849 

.44513 

52 

43 

.62257 

.41934 

.63141 

.42797 

.64010 

.43661 

.64863 

.44528 

56 

44 

.62272 

.41949 

.63156 

.42811 

.64024 

gh  Sjm 

.43676 
82° 

.64877 
Sh  35"' 

.44542 
83° 

s 

gh  23^ 

80° 

gh  27'" 

81° 

0 

45 

9.62287 

0.41963 

9.63170 

0.42825 

9.64038 

0.43690 

9.64891 

0.44557 

4 

46 

.62301 

.41977 

.63185 

.42840 

.64053 

.43704 

.64905 

.44571 

8 

47 

.62316 

.41992 

.63199 

.42854 

.64067 

.43719 

.64919 

.44586 

12 

48 

.62331 

.42006 

.63214 

.42869 

.64081 

.43733 

.64934 

.44600 

16 

49 

.62346 

.42020 

.63228 

.42883 

.64096 

.43748 

.64948 

.44614 

20 

50 

9.62361 

0.42035 

9.63243 

0.42897 

9.64110 

0.43762 

9.64962 

0.44629 

24 

51 

.62376 

.42049 

.63258 

.42912 

.64124 

.43777 

.64976 

.44643 

28 

52 

.62390 

.42063 

.63272 

.42926 

.64139 

.43791 

.64990 

.44658 

32 

53 

.62405 

.42078 

.63287 

.42941 

.64153 

.43805 

.65004 

.44672 

36 

54 

.62420 

.42092 

.63301 

.42955 

.64167 

.43820 

.65018 

.44687 

40 

55 

9.62435 

0.42106 

9.6.3316 

0.42969 

9.64181 

0.43834 

9.65032 

0.44701 

44 

56 

.62450 

.42121 

.63330 

.42984 

.64196 

.43849 

.65046 

.44716 

48 

57 

.62464 

.42135 

.63345 

.42998 

.64210 

.43863 

.65060 

.44730 

52 

58 

.62479 

.42150 

.63360 

.43013 

.64224 

.43878 

.65074 

.44745 

56 

59 

.62494 

.42164 

.63374 

.43027 

.64239 

.43892 

.65088 

.44759 

60 

60 

9.62509 

0.42178 

9.63389 

0.43041 

9.64253 

0.43907 

9.65102 

0.44774 

Table  10.    Haversine  Table 


271 


s 

5h  36"" 

84° 

5'^  40'" 

85° 

5h  44"' 

86° 

5h  48'" 

87 '^ 

Hav. 

No. 

Hav. 

No. 

Hav. 

No. 

Hav. 

9.67562 

No. 

0 

0 

9.65102 

0.44774 

9.65937 

0.45642 

9.66757 

0.46512 

0.47383 

4 

1 

.65116 

.44788 

.65950 

.45657 

.66770 

.46527 

.67576 

.47398 

8 

2 

.65130 

.44803 

.65964 

.45671 

.66784 

.46541 

.67589 

.47412 

12 

3 

.65144 

.44817 

.65978 

.45686 

.66797 

.46556 

.67602 

.47427 

16 

4 

.65158 

.44831 

.65992 

.45700 

.06811 

.46570 

.07016 

.47441 

20 

5 

9.65172 

0.44846 

9.66006 

0.45715 

9.66824 

0.46585 

9.67629 

0.47456 

24 

6 

.65186 

.44860 

.66019 

.45729 

.66838 

.46599 

.67642 

.47470 

28 

7 

.65200 

.44875 

.00033 

.45744 

.66851 

.46614 

.67656 

.47485 

32 

8 

.65214 

.44889 

.60047 

.45758 

.66865 

.46628 

.67669 

.47499 

36 

9 

.65228 

.44904 

.06061 

.45773 

.66878 

.46643 

.67682 

.47514 

40 

10 

9.65242 

0.44918 

9.66074 

0.45787 

9.66892 

0.46657 

9.67695 

0.47528 

44 

11 

.65256 

.44933 

.66088 

.45802 

.66905 

.46672 

.67709 

.47543 

48 

12 

.65270 

.44947 

.06102 

.45816 

.66919 

.46686 

.67722 

.47558 

52 

13 

.65284 

.44962 

.66116 

.45831 

.66932 

.46701 

.67735 

.47572 

56 

14 

.65298 

.44976 

.66129 

.45845 

.66946 

5h  45"' 

.46715 
86° 

.67748 

.47587 

s 

jh  g-ym 

84° 

5h  41"^ 

85° 

.■,h  /(ftii 

87° 

0 

15 

9.65312 

0.44991 

9.66143 

0.45860 

9.66959 

0.46730 

9.07702 

0.47601 

4 

16 

.65326 

.45005 

.66157 

.45874 

.66973 

.46744 

AMllb 

.47616 

8 

17 

.65340 

.45020 

.66170 

.45889 

.66986 

.46759 

.Glim 

.47630 

12 

18 

.65354 

.45034 

.66184 

.45903 

.67000 

.46773 

.07801 

.47645 

16 

19 

.65368 

.45048 

.66198 

.45918 

.07013 

.46788 

.67815 

.47659 

20 

20 

9.65382 

0.45063 

9.66212 

0.45932 

9.67027 

0.46802 

9.67828 

0.47674 

24 

21 

.65396 

.45077 

.66225 

.45947 

.67040 

.46817 

.67841 

.47688 

28 

22 

.65410 

.45092 

.66239 

.45961 

.67054 

.46831 

.67854 

.47703 

32 

23 

.65424 

.45106 

.66253 

.45976 

.67067 

.46846 

.67868 

.47717 

36 

24 

.65438 

.45121 

.66266 

.45990 

.67081 

.46860 

.67881 

.47732 

40 

25 

9.65452 

0.45135 

9.66280 

0.46005 

9.67094 

0.46875 

9.67894 

0.47746 

44 

26 

.65466 

.45150 

.66294 

.46019 

.67108 

.46890 

.67907 

.47761 

48 

27 

.65480 

.45164 

.66307 

.46034 

.67121 

.46904 

.67920 

.47775 

52 

28 

.65493 

.45179 

.66321 

.46048 

.67134 

.46919 

.67934 

.47790 

56 

29 

.65507 

.45193 

.66335 

.46063 

.67148 

.46933 

.67947 

.5/1  ^;^Q», 

.47805 
87° 

s        ' 

5h  38"" 

84° 

5h  42''' 
9.66348 

85° 
0.46077 

5''  4(>"' 
9.67161 

86° 
0.46948 

0 

30 

9.65521 

0.45208 

9.67960 

0.47819 

4 

31 

.65535 

.45222 

.66362 

.46092 

.67175 

.46962 

.67973 

.47834 

8 

32 

.65549 

.45237 

.66376 

.46106 

.67188 

.46977 

.67986 

.47848 

12 

33 

.65563 

.45251 

.66389 

.46121 

.67202 

.46991 

.68000 

.47863 

16 

34 

.65577 

.45266 

.66403 

.46135 

.67215 

.47006 

.68013 

.47877 

20 

35 

9.65591 

0.45280 

9.66417 

0.46150 

9.67228 

0.47020 

9.68026 

0.47892 

24 

36 

.65605 

.45295 

.66430 

.46164 

.67242 

.47035 

.68039 

.47906 

28 

37 

.65619 

.45309 

.66444 

.46179 

.67255 

.47049 

.68052 

.47921 

32 

38 

.65632 

.45324 

.66458 

.46193 

.67269 

.47064 

.68066 

.47935 

36 

39 

.65646 

.45338 

.66471 

.46208 

.67282 

.47078 

.68079 

.47950 

40 

40 

9.65660 

0.45353 

9.66485 

0.46222 

9.67295 

0.47093 

9.68092 

0.47964 

44 

41 

.65674 

.45367 

.66499 

.46237 

.67309 

.47107 

.68105 

.47979 

48 

42 

.65688 

.45381 

.66512 

.46251 

.67322 

.47122 

.68118 

.47993 

52 

43 

.65702 

.45396 

.66526 

.46266 

.67336 

.47136 

.68131 

.48008 

56 

44 

.65716 

.45410 

.66539 

.46280 

.67349 

5h  47'" 

.47151 
86° 

.68144 
.'V'  51'" 

.48022 

87° 

s        ' 

5h  39"^ 

84° 

5^  43"' 

85° 

0 

45 

9.65729 

0.45425 

9.66553 

0.46295 

9.67362 

0.47165 

9.68158 

0.48037 

4 

46 

.65743 

.45439 

.66567 

.46309 

.67376 

.47180 

.68171 

.48052 

8 

47 

.65757 

.45454 

.66580 

.46324 

.67389 

.47194 

.68184 

.48066 

12 

48 

.65771 

.45468 

.66594 

.46338 

.67402 

.47209 

.68197 

.48081 

16 

49 

.65785 

.45483 

.66607 

.46353 

.67416 

.47223 

.68210 

.48095 

20 

50 

9.65799 

0.45497 

9.66621 

0.46367 

9.67429 

0.47238 

9.68223 

0.48110 

24 

51 

.65812 

.45512 

.66635 

.46382 

.67443 

.47252 

.68236 

.48124 

28 

52 

.65826 

.45526 

.66648 

.46396 

.67456 

.47267 

.68249 

.48139 

32 

53 

.65840 

.45541 

.66662 

.46411 

.67469 

.47282 

.68263 

.48153 

36 

54 

.65854 

.45555 

.66675 

.46425 

.67483 

.47296 

.68276 

.48168 

40 

55 

9.65868 

0.45570 

9.66689 

0.46440 

9.67496 

0.47311 

9.68289 

0.48182 

44 

56 

.65881 

.45584 

.66702 

.46454 

.67509 

.47325 

.68302 

.48197 

48 

57 

.65895 

.45599 

.66716 

.46469 

.67522 

.47340 

.68315 

.48211 

52 

58 

.65909 

.45613 

.66730 

.46483 

.67536 

.47354 

.68328 

.48226 

56 

59 

.65923 

.45628 

.66743 

.46498 

.67549 

.47369 

.68341 

.48241 

60 

60 

9.65937 

0.45642 

9.66757 

0.46512 

9.67562 

0.47383 

9.68354 

0.48255 

272 


Table  10.    Haversine  Table 


s 

gh  g^m 

88° 

5''  56"' 

89° 

s 

Qh  Qm 

6h  4"' 

Hav. 

No. 

Hay. 

No. 

Hav. 

Hav. 

0       0 

9.68354 

0.48255 

9.69132 

0.49127 

0 

9.69897 

9.70648 

4      1 

.68367 

.48269 

.69145 

.49142 

4 

.69910 

.70661 

8       2 

.68380 

.48284 

.69158 

.49156 

8 

.69922 

.70673 

12       3 

.68393 

.48299 

.69171 

.49171 

12 

.69935 

.70686 

16       4 

.68407 

.48313 

.69184 

.49186 

16 

.69948 

.70698 

20       5 

9.68420 

0.48328 

9.09197 

0.49200 

20 

9.69960 

9.70710 

24       6 

.68433 

.48342 

.69209 

.49215 

24 

.69973 

.70723 

28       7 

.68446 

.48357 

.69222 

.49229 

28 

.69985 

.70735 

5:8   8 

.68459 

.48371 

.69235 

.49244 

32 

.69998 

.70748 

35   9 

.68472 

.48386 

.69248 

.49258 

36 

.70011 

.70760 

40     10 

9.68485 

0.48400 

9.69261 

0.49273 

40 

9.70023 

9.70772 

44  11 

.68498 

.48415 

.69274 

.49287 

44 

.70036 

.70785 

48     12 

.68511 

.48429 

.69286 

.49302 

48 

.70048 

.70797 

5:^  13 

.68524 

.48444 

.69299 

.49316 

73 

52 

.70061 

.70809 

5<?  14 

.68537 

.48459 

.69312 

.49331 

O 
> 

56 

.70074 

.70822 

s 

5h  53"" 

88° 

5h  57"" 

89° 

s 

Qh   I'n 

eh  gm 

0     15 

9.68550 

0.48473 

9.69325 

0.49346 

0 

9.70086 

9.70834 

4  16 

.68563 

.48488 

.69338 

.49360 

4 

.70099 

.70847 

8     17 

.68576 

.48502 

.69350 

.49375 

6 

8 

.70111 

.70859 

i^  18 

.68589 

.48517 

.69363 

.49389 

^ 

12 

.70124 

.70871 

16     19 

.68602 

.48531 

.69376 

.49404 

16 

.70136 

.70884 

^0  20 

9.68615 

0.48546 

9.69389 

0.49418 

20 

9.70149 

9.70896 

^4  21 

.68628 

.48560 

.69402 

.49433 

•Jl 

03 

24 

.70161 

.70908 

^5  22 

.68641 

.48575 

.69414 

.49447 

O 

28 

.70174 

.70921 

5^  23 

.68654 

.48589 

.69427 

.49462 

X;  0 ' 

32 

.70187 

.70933 

36     24 

.68667 

.48604 

.69440 

.49476 

-4—'  Q) 

36 

.70199 

.70945 

40  25 

9.68680 

0.48618 

9.69453 

0.49491 

t»  u 

40 

9.70212 

9.70958 

44  26 

.68693 

.48633 

.69465 

.49506 

44 

.70224 

.70970 

48     27 

.68706 

.48648 

.69478 

.49520 

:^^ 

48 

.70237 

.70982 

J;g  28 

.68719 

.48662 

.69491 

.49535 

Zy, 

52 

.70249 

.70995 

56     29 

.68732 

.48677 

.69504 

.49549 

0)  o 

U    >j 

56 

.70262 

.71007 

s 

5^^  54'" 

88° 

5>^  58'" 

89° 

s 

Qh  2'" 

Qh  e>n 

0  30 

9.68745 

0.48691 

9.69510 

0.49564 

0 

9.70274 

9.71019 

4  31 

.68758 

.48706 

.69529 

.49578 

4 

.70287 

.71032 

S  32 

.68771 

.48720 

.69542 

.49593 

8 

.70299 

.71044 

i^  33 

.68784 

.48735 

.69555 

.49607 

12 

.70312 

.71056 

16     34 

.68797 

.48749 

.69567 

.49622 

.■s  -^^ 

16 

.70324 

.71068 

;gO  35 

9.68810 

0.48764 

9.69580 

0.49636 

i^ 

20 

9.70337 

9.71081 

24     36 

.68823 

.48778 

.69593 

.49651 

Ss 

24 

.70349 

.71093 

^5  37 

.68836 

.48793 

.69605 

.49665 

28 

.70362 

.71105 

5^  38 

.68849 

.48807 

.69618 

.49680 

a 

_3 

32 

.70374 

.71118 

36     39 

.68862 

.48822 

.69631 

.49695 

36 

.70387 

.71130 

40  40 

9.68875 

0.48837 

9.69644 

0.49709 

"o 

40 

9.70399 

9.71142 

44    41 

.68887 

.48851 

.69656 

.49724 

^ 

44 

.70412 

.71154 

45  42 

.68900 

.48866 

.69669 

.49738 

48 

.70424 

.71167 

5;^  43 

.68913 

.48880 

.69682 

.49753 

62 

.70437 

.71179 

56     44 

.68926 

.48895 

.69694 

.49767 

1 

56 

.70449 

.71191 

s 

5h  55"" 

88° 

5h  59'" 

89° 

s 

eh  3'n 

eh  jm 

0  45 

9.68939 

0.48909 

9.69707 

0.49782 

0 

9.70462 

9.71203 

4     46 

.68952 

.48924 

.69720 

.49796 

6 

4 

.70474 

.71216 

5  47 

.68965 

.48938 

.69732 

.49811 

o 

8 

.70487 

.71228 

12     48 

.68978 

.48953 

.69745 

.49825 

12 

.70499 

.71240 

ie  49 

.68991 

.48967 

.69758 

.49840 

16 

.70512 

.71252 

;?0  50 

9.69004 

0.48982 

9.69770 

0.49855 

20 

9.70524 

9.71265 

24     51 

.69017 

.48997 

.69783 

.49869 

24 

.70537 

.71277 

;g5  52 

.69029 

.49011 

.69796 

.49884 

28 

.70549 

.71289 

32     53 

.69042 

.49026 

.69808 

.49898 

32 

.70561 

.71301 

S<5  54 

.69055 

.49040 

.69821 

.49913 

36 

.70574 

.71314 

40  55 

9.69068 

0.49055 

9.69834 

0.49927 

40 

9.70586 

9.71326 

44    56 

.69081 

.49069 

.69846 

.49942 

44 

.70599 

.71338 

45  57 

.69094 

.49084 

.69859 

.49956 

48 

.70611 

.71350 

52     58 

.69107 

.49098 

.69872 

.49971 

52 

.70624 

.71362 

56  59 

.69120 

.49113 

.69884 

.49985 

56 

.70636 

.71375 

50  60 

9.69132 

0.49127 

9.69897 

0.50000 

60 

9.70648 

9.71387 

Table  10.    Haversine  Table 


273 


s 

Qh  gm 

Qh  J^m 

Qh  iQm 

Qh  20^ 

Qh  24"^ 

Qh  28"" 

Qh   32m 

Qh  3Qm 

Hav. 

Hav. 

Hav. 

Hav. 

Hav. 

Hav. 

Hav. 

Hav. 

0 

9.71387 

9.72112 

9.72825 

9.73526 

9.74215 

9.74891 

9.75556 

9.76209 

A 

.71399 

.72124 

.72837 

.73538 

.74226 

.74902 

.75567 

.76220 

8 

.71411 

.72136 

.72849 

.73549 

.74237 

.74914 

.75578 

.76231 

12 

.71423 

.72148 

.72861 

■  .73561 

.74249 

.74925 

.75589 

.76241 

16 

.71436 

.72160 

.72873 

.73572 

.74260 

.74936 

.75600 

.76252 

20 

9.71448 

9.72172 

9.72884 

9.73584 

9.74272 

9.74947 

9.75611 

9.76263 

24 

.71460 

.72184 

.72896 

.73596 

.74283 

.74958 

.75622 

.76274 

28 

.71472 

.72196 

.72908 

.73607 

.74294 

.74969 

.75633 

.76285 

32 

.71484 

.72208 

.72920 

.73619 

.74306 

.74981 

.75644 

.76296 

36 

.71496 

.72220 

.72931 

.73630 

.74317 

.74992 

.75655 

.76306 

40 

9.71509 

9.72232 

9.72943 

9.73642 

9.74328 

9.75003 

9.75666 

9.76317 

44 

.71521 

.72244 

.72955 

.73653 

.74340 

.75014 

.75677 

.76328 

48 

.71533 

.72256 

.72967 

.73665 

.74351 

.75025 

.75688 

.76338 

62 

.71545 

.72268 

.72978 

.73676 

.74362 

.75036 

.75698 

.76349 

66 

.71557 

.72280 

.72990 

.73688 

.74374 

.75047 

.75709 

.76360 

s 

^h  gm 

Qh  igm 

Qh  Jjm 

Qh  21^ 

Qh  25'^ 

Qh  29"' 

Qh  33m 

Qh   37m 

0 

9.71569 

9.72292 

9.73002 

9.73699 

9.74385 

9.75059 

9.75720 

9.76371 

4 

.71582 

.72304 

.73014 

.73711 

.74396 

.75070 

.75731 

.76381 

8 

.71594 

.72316 

.73025 

.73722 

.74408 

.75081 

.75742 

.76392 

12 

.71606 

.72328 

.73037 

.73734 

.74419 

.75092 

.75753 

.76403 

16 

.71618 

.72340 

.73049 

.73746 

.74430 

.75103 

.75764 

.76414 

20 

9.71630 

9.72352 

9.73060 

9.73757 

9.74442 

9.75114 

9.75775 

9.76424 

24 

.71642 

.72363 

.73072 

.73769 

.74453 

.75125 

.75786 

.76435 

28 

.71654 

.72375 

.73084 

.73780 

.74464 

.75136 

.75797 

.76446 

32 

.71666 

.72387 

.73096 

.73792 

.74475 

.75147 

.75808 

.76456 

36 

.71679 

.72399 

.73107 

.73803 

.74487 

.75159 

.75819 

.76467 

40 

9.71691 

9.72411 

9.73119 

9.73815 

9.74498 

9.75170 

9.75830 

9.76478 

44 

.71703 

.72423 

.73131 

.73826 

.74509 

.75181 

.75840 

.76489 

48 

.71715 

.72435 

.73142 

.73838 

.74521 

.75192 

.75851 

.76499 

62 

.71727 

.72447 

.73154 

.73849 

.74532 

.75203 

.75862 

.76510 

66 

.71739 

.72459 

.73166 

.73860 

.74543 

.75214 

.75873 
Qh  34"^ 

.76521 

Qh  3Sm 

s 

Q-h  10"" 

Qh  Ijni 

Qh  IS"" 

Qh  22"^ 

Qh  26"' 

Qh  30m 

0 

9.71751 

9.72471 

9.73177 

9.73872 

9.74554 

9.75225 

9.75884 

9.76531 

4 

.71763 

.72482 

.73189 

.73883 

.74566 

.75236 

.75895 

.76542 

8 

.71775 

.72494 

.73201 

.73895 

.74577 

.75247 

.75906 

.76553 

12 

.71787 

.72506 

.73212 

.73906 

.74588 

.75258 

.75917 

.76563 

16 

.71800 

.72518 

.73224 

.73918 

.74600 

.75269 

.75927 

.76574 

20 

9.71812 

9.72530 

9.73236 

9.73929 

9.74611 

9.75280 

9.75938 

9.76585 

24 

.71824 

.72542 

.73247 

.73941 

.74622 

.75291 

.75949 

.76595 

28 

.71836 

.72554 

.73259 

.73952 

.74633 

.75303 

.75960 

.76606 

32 

.71848 

.72565 

.73271 

.73964 

.74645 

.75314 

.75971 

.76617 

36 

.71860 

.72577 

.73282 

.73975 

.74656 

.75325 

.75982 

.76627 

40 

9.71872 

9.72589 

9.73294 

9.73987 

9.74667 

9.75336 

9.75993 

9.76638 

44 

.71884 

.72601 

.73306 

.73998 

.74678 

.75347 

.76004 

.76649 

48 

.71896 

.72613 

.73317 

.74009 

.74690 

.75358 

.76014 

.76659 

62 

.71908 

.72625 

.73329 

.74021 

.74701 

.75369 

.76025 

.76670 

66 

.71920 

.72637 

.73341 

.74032 

.74712 

.75380 

.76036 

.76681 

s 

gh  ipn 

Qh  15m 

Qh  jgm 

Qh  23"^ 

Qh  27"" 

Qh  3im 

6h  36"" 

Qh  39m 

0 

9.71932 

9.72648 

9.73352 

9.74044 

9.74723 

9.75391 

9.76047 

9.76691 

4 

.71944 

.72660 

.73364 

.74055 

.74734 

.75402 

.76058 

.76702 

8 

.71956 

.72672 

.73375 

.74067 

.lAim 

.75413 

.76069 

.76713 

12 

.71968 

.72684 

.73387 

.74078 

.74757 

.75424 

.76079 

.76723 

16 

.71980 

.72696 

.73399 

.74089 

.74768 

.75435 

.76090 

.76734 

20 

9.71992 

9.72708 

9.73410 

9.74101 

9.74779 

9.75446 

9.76101 

9.76745 

24 

.72004 

.72719 

.73422 

.74112 

.74791 

.75457 

.76112 

.76755 

28 

.72016 

.72731 

.73433 

.74124 

.74802 

.75468 

.76123 

.76766 

32 

.72028 

.72743 

.73445 

.74135 

.74813 

.75479 

.76134 

.76777 

36 

.72040 

.72755 

.73457 

.74146 

.74824 

.75490 

.76144 

.76787 

40 

9.72052 

9.72767 

9.73468 

9.74158 

9.74835 

9.75501 

9.76155 

9.76798 

44 

.72064 

.72778 

.73480 

.74169 

.74846 

.75512 

.76166 

.76808 

48 

.72076 

.72790 

.73491 

.74181 

.74858 

.75523 

.76177 

.76819 

62 

.72088 

.72802 

.73503 

.74192 

.74869 

.75534 

.76188 

.76830 

66 

.72100 

.72814 

.73515 

.74203 

.74880 

.75545 

.76198 

.76840 

60 

9.72112 

9.72825 

9.73526 

9.74215 

9.74891 

9.75556 

9.76209 

9.76851 

274 


Table  10.    Hayersine  Table 


s 

6h  40'" 

6h  44'^ 

6h  48"" 
Hav. 

eh   .5^m 

Hav. 

6h  56"" 

7h  0'" 

yh  jm 

7A  gm 

Hav. 

Hav. 

Hav. 

9.79306 

Hav. 

9.79893 

Hav. 

Hav. 

0 

9.76851 

9.77481 

9.78101 

9.78709 

9.80470 

9.81036 

4 

.76861 

.77492 

.78111 

.78719 

.79316 

.79903 

.80479 

.81045 

8 

.76872 

.77502 

.78121 

.78729 

.79326 

.79913 

.80489 

.81054 

12 

.76883 

.77512 

.78131 

.78739  ■ 

.79336 

.79922 

.80498 

.81064 

16 

.76893 

.77523 

.78141 

.78749 

.79346 

.79932 

.80508 

.81073 

20 

9.76904 

9.77533 

9.78152 

9.78759 

9.79356 

9.79942 

9.80517 

9.81082 

24 

.76914 

.77544 

.78162 

.78769 

.79366 

.79951 

.80527 

.81092 

28 

.76925 

.77554 

.78172 

.78779 

.79376 

.79961 

.80536 

.81101 

32 

.76936 

.77564 

.78182 

.78789 

.79385 

.79971 

.80546 

.81110 

36 

.76946 

.77575 

.78192 

.78799 

.79395 

.79980 

.80555 

.81120 

40 

9.76957 

9.77585 

9.78203 

9.78809 

9.79405 

9.79990 

9.80565 

9.88129 

U 

.76967 

.77596 

.78213 

.78819 

.79415 

.80000 

.80574 

.81138 

48 

.76978 

.77606 

.78223 

.78829 

.79425 

.80009 

.80584 

.81148 

62 

.76988 

.77616 

.78233 

.78839 

.79434 

.80019 

.80593 

.81157 

56 

.76999 

6^  41"^ 

.77627 

(jh  45^ 

.78243 

.78849 

.79444 

.80029 

.80603 

.81166 

8 

6h  45"» 

6h  53"' 

eh  r^jm 

-h    J>n 

'^h  5m 

7  A  <j>n 

0 

9.77009 

9.77637 

9.78254 

9.78859 

9.79454 

9.80038 

9.80612 

9.81176 

4 

.77020 

.77647 

.78264 

.78869 

.79464 

.80048 

.80622 

.81185 

8 

.77031 

.77658 

.78274 

.78879 

.79474 

.80058 

.80631 

.81194 

12 

.77041 

.77668 

.78284 

.78889 

.79484 

.80067 

.80641 

.81204 

16 

.77052 

.77679 

.78294 

.78899 

.79493 

.80077 

.80650 

.81213 

20 

9.77062 

9.77689 

9.78305 

9.78909 

9.79503 

9.80087 

9.80660 

9.81222 

24 

.77073 

.77699 

.78315 

.78919 

.79513 

.80096 

.80669 

.81231 

28 

.77083 

.77710 

.78325 

.78929 

.79523 

.80106 

.80678 

.81241 

32 

.77094 

.77720 

.78335 

.78939 

.79533 

.80116 

.80688 

.81250 

36 

.77104 

.77730 

.78345 

.78949 

.79542 

.80125 

.80697 

.81259 

40 

9.77115 

9.77741 

9.78355 

9.78959 

9.79552 

9.80135 

9.80707 

9.81269 

U 

.77125 

.77751 

.78365 

.78969 

.79562 

.80144 

.80716 

.81278 

48 

.77136 

.77761 

.78376 

.78979 

.79572 

.80154 

.80726 

.81287 

52 

.77146 

.77772 

.78386 

.78989 

.79582 

.80164 

.80735 

.81296 

56 

.77157 

.77782 

.78396 

.78999 

.79591 

.80173 

.80745 

.81306 

s 

6h  j^2"' 

6^^  46"" 

Qh  gO"^ 

6h  54"^ 

6^^  58"" 

Jh  2"' 

jh  e»' 

jh  10"' 

0 

9.77167 

9.77792 

9.78406 

9.79009 

9.79601 

9.80183 

9.80754 

9.81315 

4 

.77178 

.77803 

.78416 

.79019 

.79611 

.80192 

.80763 

.81324 

8 

.77188 

.77813 

.78426 

.79029 

.79621 

.80202 

.80773 

.81333 

12 

.77199 

.77823 

.78436 

.79039 

.79631 

.80212 

.80782 

.81343 

16 

.77209 

.77834 

.78447 

.79049 

.79640 

.80221 

.80792 

.81352 

20 

9.77220 

9.77844 

9.78457 

9.79059 

9.79650 

9.80231 

9.80801 

9.81361 

24 

.77230 

.77854 

.78467 

.79069 

.79660 

.80240 

.80811 

.81370 

28 

.77241 

.77864 

.78477 

.79079 

.79670 

.80250 

.80820 

.81380 

32 

.77251 

.77875 

.78487 

.79089 

.79679 

.80260 

.80829 

.81389 

36 

.77262 

.77885 

.78497 

.79099 

.79689 

.80269 

.80839 

.81398 

40 

9.77272 

9.77895 

9.78507 

9.79108 

9.79699 

9.80279 

9.80848 

9.81407 

U 

.77283 

.77906 

.78517 

.79118 

.79709 

.80288 

.80858 

.81417 

48 

.77293 

.77916 

.78528 

.79128 

.79718 

.80298 

.80867 

.81426 

52 

.77304 

.77926 

.78538 

.79138 

.79728 

.80307 

.80876 

.81435 

56 

.77314 

6h  43"" 

.77936 

eh  47'« 

.78548 
eh  gpn 

.79148 

6h  55"^ 

.79738 

.80317 

.80886 

.81444 

s 

Of'  59"' 

7/1  s"' 

7h  jm 

7/1  11^ 

0 

9.77325 

9.77947 

9.78558 

9.79158 

9.79748 

9.80327 

9.80895 

9.81454 

4 

.77335 

.77957 

.78568 

.79168 

.79757 

.80336 

.80905 

.81463 

8 

.77346 

.77967 

.78578 

.79178 

.79767 

.80346 

.80914 

.81472 

12 

.77356 

.77978 

.78588 

.79188 

.79777 

.80355 

.80923 

.81481 

16 

.77366 

.77988 

.78598 

.79198 

.79787 

.80365 

.80933 

.81490 

20 

9.77377 

9.77998 

9.78608 

9.79208 

9.79796 

9.80374 

9.80942 

9.81500 

24 

.77387 

.78008 

.78618 

.79217 

.79806 

.80384 

.80952 

.81509 

28 

.77398 

.78019 

.78628 

.79227 

.79816 

.80393 

.80961 

.81518 

32 

.77408 

.78029 

.78638 

.79237 

.79825 

.80403 

.80970 

.81527 

36 

.77419 

.78039 

.78649 

.79247 

.79835 

.80413 

.80980 

.81536 

40 

9.77429 

9.78049 

9.78659 

9.79257 

9.79845 

9.80422 

9.80989 

9.81546 

U 

.77440 

.78060 

.78669 

.79267 

.79855 

.80432 

.80998 

.81555 

48 

.77450 

.78070 

.78679 

.79277 

.79864 

.80441 

.81008 

.81564 

52 

.77460 

.78080 

.78689 

.79287 

.79874 

.80451 

.81017 

.81573 

56 

.77471 

.78090 

.78699 

.79297 

.79884 

.80460 

.81026 

.81582 

60 

9.77481 

9.78101 

9.78709 

9.79306 

9.79893 

9.80470 

9.81036 

9.81592 

Table  10.    Haversine  Table 


275 


s 

7^  12"' 

7h  10"' 

-h  20"' 

7''  24"' 

7''  28'" 

7/1  32'" 

7^  s6'" 

7h  40"^ 

Hav. 

Hav. 

Hav. 

Hav. 

Hav. 

Hav. 

Hav. 

9^84718 

Hav. 

0 

9^81592 

9.82137 

9.82673 

9.83199 

9^83715 

9.84221 

9.85206 

4 

.81601 

■82146 

■82682 

■83207 

■83723 

■84230 

■84726 

.85214 

8 

■81610 

.82155 

■82691 

■83216 

■83732 

■84238 

■84735 

.85222 

12 

■81619 

.82164 

■82699 

.83225 

■83740 

■84246 

■84743 

.85230 

16 

■81628 

.82173 

■82708 

■83233 

.83749 

.84255 

.84751 

.85238 

20 

9^81637 

9.82182 

9^82717 

9^83242 

9.83757 

9.84263 

9.84759 

9.85246 

24 

.81647 

.82191 

■82726 

■83251 

.83766 

.84271 

.84767 

.85254 

28 

.81656 

■82200 

■82735 

■83259 

.83774 

.84280 

.84776 

.85262 

32 

.81665 

.82209 

■82744 

.83268 

.83783 

.84288 

.84784 

.85270 

36 

.81674 

.82218 

■82752 

.83277 

.83791 

.84296 

.84792 

.85278 

40 

9.81683 

9.82227 

9.82761 

9.83285 

9.83800 

9.84305 

9.84800 

9.85286 

44 

■81692 

.82236 

.82770 

.83294 

.83808 

■84313 

.84808 

.85294 

48 

■81701 

■82245 

■82779 

.83303 

.83817 

■84321 

.84817 

.85302 

52 

■81711 

■82254 

.82788 

.83311 

.83825 

■84330 

.84825 

.85310 

56 

■81720 

■82263 

.82796 
7^  2V" 

.83320 

jh  25'" 

.83834 

7h  29"' 

.84338 
jh  S3'" 

.84833 

.85318 

s 

7h  13"' 

Th   Ijm 

7^  37"' 

7A  J^im 

0 

9^81729 

9^82272 

9.82805 

9.83329 

9.83842 

9.84340 

9.84841 

9.85326 

4 

■81738 

■82281 

■82814 

■83337 

■83851 

■84355 

.84849 

.85334 

8 

■81747 

.82290 

■82823 

■83346 

■83859 

.84363 

.84857 

.85342 

12 

.81756 

.82299 

■82832 

.83355 

■83868 

.84371 

.84866 

.85350 

16 

.81765 

■82308 

.82840 

.83363 

■83876 

.84380 

.84874 

.85358 

20 

9.81775 

9.82317 

9.82849 

9.83372 

9^83885 

9.84388 

9.84882 

9.85366 

24 

.81784 

■82326 

■82858 

.83380 

■83893 

.84396 

.84890 

.85374 

28 

.81793 

■82335 

■82867 

.83389 

■83902 

.84405 

.84898 

.85382 

32 

.81802 

■82344 

■82876 

■83398 

■83910 

.84413 

.84906 

.85390 

36 

.81811 

■82353 

■82884 

■83406 

.83919 

.84421 

■84914 

.85398 

40 

9.81820 

9^82362 

9^82893 

9^83415 

9.83927 

9.84430 

9^84923 

9.85406 

44 

.81829 

.82371 

.82902 

■83424 

■83935 

.84438 

■84931 

.85414 

48 

.81838 

.82380 

.82911 

■83432 

■83944 

.84446 

■84939 

.85422 

52 

.81847 

.82388 

.82920 

■83441 

■83952 

.84454 

■84947 

.85430 

56 

■81857 

.82397 

■82928 

.83449 

.83961 

.84463 

■84955 

.85438 

s 

7A  IJ^m 

jh  is>n 

jh  22"^ 

7A  26'" 

jh  30'" 

7h  34'" 

7^  38"' 

jh  ^2"' 

0 

9^81866 

9.82406 

9.82937 

9.83458 

).  83969 

9.84471 

9^84963 

9.85446 

4 

■81875 

.82415 

.82946 

.83467 

■83978 

.84479 

■84971 

■85454 

8 

.81884 

.82424 

.82955 

.83475 

.83986 

.84488 

■84979 

.85462 

12 

.81893 

.82433 

.82963 

.83484 

.83995 

.84496 

.84988 

.85470 

16 

.81902 

.82442 

.82972 

.83492 

.84003 

.84504 

.84996 

.85478 

20 

9.81911 

9.82451 

9.82981 

9.83501 

:).84011 

9.84512 

9.85004 

9.85486 

24 

■81920 

.82460 

.82990 

.83510 

■84020 

.84521 

.85012 

.85494 

28 

■81929 

■82469 

.82998 

.83518 

■84028 

.84529 

.85020 

.85502 

32 

.81938 

■82478 

.83007 

.83527 

■84037 

.84537 

.85028 

.85510 

36 

.81947 

.82487 

.83016 

.83535 

.84045 

.84545 

.85036 

.85518 

40 

9.81956 

9.82495 

9.83025 

9.83544 

9.84054 

9.84554 

9.85044 

9.85526 

44 

.81965 

.82504 

.83033 

.83552 

.84062 

.84562 

.85052 

.85534 

48 

.81975 

.82513 

.83042 

.83561 

.84070 

.84570 

.85061 

.85542 

52 

.81984 

.82522 

■83051 

.83570 

.84079 

.84578 

.85069 

.85550 

56 

.81993 

■82531 

■83059 

.83578 

.84087 

.84587 

.85077 

.85557 

s 

7h  15"' 

jh  igm 

7^  23'" 

7''  27'" 

7h  31'" 

jh  So"' 

7^  39"' 

7/1  43m 

0 

9.82002 

9.82540 

9^83068 

9.83587 

9.84096 

9.84595 

9.85085 

9.85565 

4 

.82011 

.82549 

■83077 

.83595 

.84104 

.84603 

.85093 

.85573 

8 

.82020 

.82558 

■83086 

.83604 

.84112 

.84611 

■85101 

.85581 

12 

.82029 

.82567 

■83094 

.83612 

.84121 

.84620 

.85109 

.85589 

16 

.82038 

.82575 

■83103 

.83621 

.84129 

.84628 

.85117 

.85597 

20 

9.82047 

9.82584 

9^83112 

9.83630 

9.84138 

9.84636 

9.85125 

9.85605 

24 

.82056 

.82593 

.83120 

.83638 

.84146 

.84644 

.85133 

.85613 

28 

.82065 

.82602 

.83129 

.83647 

.84154 

.84653 

.85141 

.85621 

32 

.82074 

.82611 

.83138 

.83655 

.84163 

■84661 

.85149 

.85629 

36 

.82083 

.82620 

.83147 

.83664 

.84171 

.84669 

.85158 

.85637 

40 

9.82092 

9.82629 

9.83155 

9.83672 

9.84179 

9.84677 

9.85166 

9.85645 

44 

■82101 

■82638 

.83164 

.83681 

.84188 

.84685 

.85174 

.85653 

48 

.82110 

■82646 

.83173 

.83689 

.84196 

.84694 

.85182 

.85660 

52 

.82119 

■82655 

.83181 

.83698 

■84205 

.84702 

.85190 

.85668 

56 

.82128 

■82664 

.83190 

■83706 

.84213 

.84710 

.85198 

.85676 

60 

9.82137 

9.82673 

9.83199 

9^83715 

9.84221 

9.84718 

9.85206 

9.85684 

276 


Table  10.    Hayersine  Table 


s 

7h  44"^ 

771  j^8^ 

7h  52'" 

7h  56"" 

gh  Qm 

gh  ^m 

gh  gm 

gh  12"" 

Hav. 

Hav. 

Hav. 

Hav. 

Hav. 

9.87506 

Hav. 

9.87939 

Hav. 

Hav. 

0 

9.85684 

9.86153 

9.86613 

9.87064 

9.88364 

9.88780 

4 

.85692 

.86161 

.86621 

.87072 

.87513 

.87947 

.88371 

.88787 

8 

.85700 

.86169 

.86628 

.87079 

.87521 

.87954 

.88378 

.88793 

12 

.85708 

.86176 

.86636 

.87086 

.87528 

.87961 

.88385 

.88800 

16 

.85716 

.86184 

.86643 

.87094 

.87535 

.87968 

.88392 

.88807 

20 

9.85724 

9.86192 

9.86651 

9.87101 

9.87543 

9.87975 

9.88399 

9.88814 

24 

.85731 

.86200 

.86659 

.87109 

.87550 

.87982 

.88406 

.88821 

28 

.85739 

.86207 

.86666 

.87116 

.87557 

.87989 

.88413 

.88828 

82 

.85747 

.86215 

.86674 

.87124 

.87564 

.87996 

.88420 

.88835 

'36 

.85755 

.86223 

.86681 

.87131 

.87572 

.88004 

.88427 

.88841 

40 

9.85763 

9.86230 

9.86689 

9.87138 

9.87579 

9.88011 

9.88434 

9.88848 

44 

.85771 

.86238 

.86696 

.87146 

.87586 

.88018 

.88441 

.88855 

48 

.85779 

.86246 

.86704 

.87153 

.87593 

.88025 

.88448 

.88862 

62 

.85787 

.86254 

.86712 

.87161 

.87601 

.88032 

.88455 

.88869 

56 

.85794 

.86261 

.86719 

.87168 

.87608 

.88039 

.88462 

.88876 

s  ^ 

7h  J^5"' 

7h  A9'" 

7A  ggm 

Jh  g^m 

Sh   I'n 

gh   jm 

gh  gm 

gh  23"' 

0 

9.85802 

9.86269 

9.86727 

9.87175 

9.87615 

9.88046 

9.88469 

9.88882 

4 

.85810 

.86277 

.86734 

.87183 

.87623 

.88053 

.88476 

.88889 

8 

.85818 

.86284 

.86742 

.87190 

.87630 

.88061 

.88483 

.88896 

12 

.85826 

.86292 

.86749 

.87198 

.87637 

.88068 

.88490 

.88903 

16 

.85834 

.86300 

.86757 

.87205 

.87644 

.88075 

.88496 

.88910 

20 

9.85841 

9.86307 

9.86764 

9.87212 

9.87652 

9.88082 

9.88503 

9.88916 

24 

.85849 

.86315 

.86772 

.87220 

.87659 

.88089 

.88510 

.88923 

28 

.85857 

.86323 

.86780 

.87227 

.87666 

.88096 

.88517 

.88930 

32 

.85865 

.86331 

.86787 

.87235 

.87673 

.88103 

.88524 

.88937 

36 

.85873 

.86338 

.86795 

.87242 

.87680 

.88110 

.88531 

.88944 

40 

9.85881 

9.86346 

9.86802 

9.87249 

9.87688 

9.88117 

9.88528 

9.88950 

44 

.85888 

.86354 

.86810 

.87257 

.87695 

.88124 

.88545 

.88957 

48 

.85896 

.86361 

.86817 

.87264 

.87702 

.88131 

.88552 

.88964 

62 

.85904 

.86369 

.86825 

.87271 

.87709 

.88139 

.88559 

.88971 

66 

.85912 

.86377 

.86832 

.87279 

.87717 

gh  ^m 

.88146 

gh  gm 

.88566 

.88978 

s 

7h   46"n 

7h  50"^ 

r'l  54"^ 

7/1  ss"" 

gh  20'"' 

8'^  14"" 

0 

9.85920 

9.86384 

9.86840 

9.87286 

9.87724 

9.88153 

9.88573 

9.88984 

4 

.85928 

.86392 

.86847 

.87294 

.87731 

.88160 

.88580 

.88991 

8 

.85935 

.86400 

.86855 

.87301 

.87738 

.88167 

.88587 

.88998 

12 

.85943 

.86407 

.86862 

.87308 

.87745 

.88174 

.88594 

.89005 

16 

.85951 

.86415 

.86870 

.87316 

.87753 

.88181 

.88600 

.89012 

20 

9.85959 

9.86423 

9.86877 

9.87323  ■ 

9.87760 

9.88188 

9.88607 

9.89018 

24 

.85967 

.86430 

.86885 

.87330 

.87767 

.88195 

.88614 

.89025 

28 

.85974 

.86438 

.86892 

.87338 

.87774 

.88202 

.88621 

.89032 

32 

.85982 

.86446 

.86900 

.87345 

.87782 

.88209 

.88628 

.89039 

36 

.85990 

.86453 

.86907 

.87352 

.87789 

.88216 

.88635 

.89045 

40 

9.85998 

9.86461 

9.86915 

9.87360 

9.87796 

9.88223 

9.88642 

9.89052 

44 

.86006 

.86468 

.86922 

.87367 

.87803 

.88230 

.88649 

.89059 

48 

.86013 

.86476 

.86.930 

.87374 

.87810 

.88237 

.88656 

.89066 

52 

.86021 

.86484 

.86937 

.87382 

.87818 

.88244 

.88663 

.89072 

56 

.86029 

.86491 

.86945 

.87389 

.87825 

.88252 

.88670 

.89079 

s 

7h  J^jm 

7h  61'" 

7A  ggm 

7/1  ggm 

gh  gm 

gh'ym 

gh  iim 

8^  15"" 

0 

9.86037 

9.86499 

9.86952 

9.87396 

9.87832 

9.88259 

9.88677 

9.89086 

4 

.86045 

.86507 

.86960 

.87404 

.87839 

.88266 

.88683 

.89093 

8 

.86052 

.86514 

.86967 

.87411 

.87846 

.88273 

.88690 

.89099 

12 

.86060 

.86522 

.86975 

.87418 

.87853 

.88280 

.88697 

.89106 

16 

.86068 

.86529 

.86982 

.87426 

.87861 

.88287 

.88704 

.89113 

20 

9.86076 

9.86537 

9.86990 

9.87433 

9.87868 

9.88294 

9.88711 

9.89120 

24 

.86083 

.86545 

.86997 

.87440 

.87875 

.88301 

.88718 

.89126 

28 

.86091 

.86552 

.87004 

.87448 

.87882 

.88308 

.88725 

.89133 

32 

.86099 

.86560 

.87012 

.87455 

.87889 

.88315 

.88732 

.89140 

36 

.86107 

.86568 

.87019 

.87462 

.87896 

.88322 

.88739 

.89147 

40 

9.86114 

9.86575 

9.87027 

9.87470 

9.87904 

9.88329 

9.88745 

9.89153 

44 

.86122 

.86583 

.87034 

.87477 

.87911 

.88336 

.88752 

.89160 

48 

.86130 

.86590 

.87042 

.87484 

.87918 

.88343 

.88759 

.89167 

62 

.86138 

.86598 

.87049 

.87492 

.87925 

.88350 

.88766 

.89174 

66 

.86145 

.86606 

.87057 

.87499 

.87932 

.88357 

.88773 

.89180 

60 

9.86153 

9.86613 

9.87064 

9.87506 

9.87939 

9.88364 

9.88780 

9.89187 

Table  10.    Haversine  Table 


277 


s 

gh  IQm 

gh  20'" 

8''  24"' 

8h  28"' 

gh  S2'" 

8^  36'" 

8h  40"' 

8^  44"^ 

Hav. 

Hav. 

Hav. 

Hav. 

Hav. 

Hav. 

Hav. 

Hav. 

0 

9.89187 

9.89580 

9.89976 

9.90358 

9.90732 

9.91098 

9.91455 

9.91805 

4 

.89194 

.89592 

.89983 

.90365 

.90738 

.91104 

.91461 

.91810 

8 

.89200 

.89599 

.89989 

.90371 

.90744 

.91110 

.91467 

.91816 

12 

.89207 

.89006 

.89995 

.90377 

.90751 

.91116 

.91473 

.91822 

16 

.89214 

.89612 

.90002 

.90383 

.90757 

.91122 

.91479 

.91828 

20 

9.89221 

9.89619 

9.90008 

9.90390 

9.90763 

9.91128 

9.91485 

9.91833 

u 

.89227 

.89625 

.90015 

.90396 

.90769 

.91134 

.91490 

.91839 

28 

.89234 

.89632 

.90021 

.90402 

.90775 

.91140 

.91496 

.91845 

32 

.89241 

.89638 

.90028 

.90409 

.90781 

.91146 

.91502 

.91851 

36 

.89247 

.89645 

.90034 

.90415 

.90787 

.91152 

.91508 

.91856 

40 

9.89254 

9.89651 

9.90040 

9.90421 

9.90794 

9.91158 

9.91514 

9.91862 

U 

.89261 

.89658 

.90047 

.90428 

.90800 

.91164 

.91520 

.91868 

48 

.89267 

.89665 

.90053 

.90434 

.90806 

.91170 

.91526 

.91874 

52 

.89274 

.89671 

.90060 

.90440 

.90812 

.91176 

.91532 

.91879 

56 

.89281 

.89678 

.90066 

.90446 

.90818 

.91182 

.91537 

.91885 

s 

gh  17'n 

gh  21"' 

gh  25"^ 

gh  2,9" 

gh  S3"' 

gh  S7'" 

gh  ^im 

8f'  4-5"' 

0 

9.89287 

9.89684 

9.90072 

9.90452 

9.90824 

9.91188 

9.91543 

9.91891 

4 

.89294 

.80691 

.90079 

.90459 

.90830 

.91194 

.91549 

.91896 

8 

.89301 

.89697 

.90085 

.90465 

.90836 

.91200 

.91555 

.91902 

12 

.89308 

.89704 

.90092 

.90471 

.90843 

.91206 

.91561 

.91908 

16 

.89314 

.89710 

.90098 

.90478 

.90849 

.91212 

.91567 

.91914 

20 

9.89321 

9.89717 

9.90104 

9.90484 

9.90855 

9.91218 

9.91573 

9.91919 

24 

.89328 

.89723 

.90111 

.90490 

.90861 

.91224 

.91578 

.91925 

28 

.89334 

.89730 

.90117 

.90496 

.90867 

.91230 

.91584 

.91931 

32 

.89341 

.89736 

.90124 

.90503 

.90873 

.91236 

.91590 

.91936 

36 

.89348 

.89743 

.90130 

.90509 

.90879 

.91242 

.91596 

.91942 

40 

9.89354 

9.89749 

9.90136 

9.90515 

9.90885 

9.91248 

9.91602 

9.91948 

44 

.89361 

.89756 

.90143 

.90521 

.90892 

.91254 

.91608 

.91954 

48 

.89368 

.89763 

.90149 

.90527 

.90898 

.91260 

.91613 

.91959 

52 

.89374 

.89769 

.90156 

.90534 

.90904 

.91265 

.91619 

.91965 

56 

.89381 

.89776 

.90162 

.90540 

.90910 

.91271 

.91625 

.91971 

s 

8h  18"" 

gh  22^ 

gh  26'" 

Sh  30'" 

8h  34"' 

gh  Jgm 

8h  42'" 

8^  46"' 

0 

9.89387 

9.89782 

9.90168 

9.90546 

9.90916 

9.91277 

9.91631 

9.91976 

4 

.89394 

.89789 

.90175 

.90552 

.90922 

.91283 

.91637 

.91982 

8 

.89400 

.89795 

.90181 

.90559 

.90928 

.91289 

.91643 

.91988 

12 

.89407 

.89802 

.90187 

.90565 

.90934 

.91295 

.91648 

.91993 

16 

.89414 

.89808 

.90194 

.90571 

.90940 

.91301 

.91654 

.91999 

20 

9.89421 

9.89815 

9.90200 

9.90577 

9.90946 

9.91307 

9.91660 

9.92005 

24 

.89427 

.89821 

.90206 

.90584 

.90952 

.91313 

.91666 

.92010 

28 

.89434 

.89828 

.90213 

.90590 

.90958 

.91319 

.91672 

.92016 

32 

.89441 

.89834 

.90219 

.90596 

.90905 

.91325 

.91677 

.92022 

36 

.89447 

.89840 

.90225 

.90602 

.90971 

.91331 

.91683 

.92027 

40 

9.89454 

9.89847 

9.90232 

9.90608 

9.90977 

9.91337 

9.91689 

9.92033 

U 

.89460 

.89853 

.90238 

.90615 

.90983 

.91343 

.91695 

.92039 

48 

.89467 

.89860 

.90244 

.90621 

.90989 

.91349 

.91701 

.92044 

52 

.89474 

.89866 

.90251 

.90627 

.90995 

.91355 

.91706 

.92050 

56 

.89480 

.89873 

.90257 

.90633 

.91001 

.91361 

.91712 

.92056 

s 

gh  jgm 

gh  23''' 

gh  27"' 

gh  Sl'n 

8h  So'" 

gh  SO"" 

8''  43"^ 

gh  J^jm 

0 

9.89487 

9.89879 

9.90264 

9.90639 

9.91007 

9.91367 

9.91718 

9.92061 

4 

.89493 

.89886 

.90270 

.90646 

.91013 

.91372 

.91724 

.92067 

8 

.89500 

.89892 

.90276 

.90652 

.91019 

.91378 

.91730 

.92073 

12 

.89507 

.89899 

.90282 

.90658 

.91025 

.91384 

.91735 

.92078 

16 

.89513 

.89905 

.90289 

.90664 

.91031 

.91390 

.91741 

.92084 

20 

9.89520 

9.89912 

9.90295 

9.90670 

9.91037 

9.91396 

9.91747 

9.92090 

24 

.89527 

.89918 

.90301 

.90676 

.91043 

.91402 

.91753 

.92095 

28 

.89533 

.89925 

.90308 

.90683 

.91049 

.91408 

.91758 

.92101 

32 

.89540 

.89931 

.90314 

.90689 

.91055 

.91414 

.91764 

.92107 

36 

.89546 

.89938 

.90320 

.90695 

.91061 

.91420 

.91770 

.92112 

40 

9.89553 

9.89944 

9.90327 

9.90701 

9.01067 

9.91426 

9.91776 

9.92118 

44 

.89559 

.89950 

.90333 

.90707 

.91074 

.91432 

.91782 

.92124 

48 

.89566 

.89957 

.90339 

.90714 

.91080 

.91437 

.91787 

.92129 

52 

.89573 

.89963 

.90346 

.90720 

.91086 

.91443 

.91793 

.92135 

56 

.89579 

.89970 

.90352 

.90726 

.91092 

.91449 

.91799 

.92140 

60 

9.89586 

9.89976 

9.90358 

9.90732 

9.19098 

9.91455 

9.91805 

9.92146 

278 


Table  10.    Hayersine  Table 


s 

8h  48"" 

gh  52"" 

8h  56"^ 

QhQm 

Qh  jm 

Qh  8"^ 

Qh  12m. 

Qh  igm 

Hav. 

Hav. 

Hav. 

9.92805 

Hav. 

Hav. 

Hav. 

Hav. 

Hav. 

0 

9.92146 

9.92480 

9.93123 

9.93433 

9.93736 

9.94030 

9.94318 

4 

.92152 

.92485 

.92811 

.93128 

.93438 

.93741 

.94035 

.94322 

8 

.92157 

.92491 

.92816 

.93134 

.93443 

.93746 

.94040 

.94327 

12 

.92163 

.92496 

.92821 

.93139 

.93448 

.93751 

.94045 

.94332 

16 

.92169 

.92502 

.92827 

.93144 

.93454 

.93755 

.94050 

.94336 

20 

9.92174 

9.92507 

9.92832 

9.93149 

9.93459 

9.93760 

9.94055 

9.94341 

24 

.92180 

.92512 

.92837 

.93154 

.93464 

.93765 

.94059 

.94346 

28 

.92185 

.92518 

.92843 

.93160 

.93469 

.93770 

.94064 

.94351 

32 

.92191 

.92523 

.92848 

.93165 

.93474 

.93775 

.94069 

.94355 

36 

.92197 

.92529 

.92853 

.93170 

.93479 

.93780 

.94074 

.94360 

40 

9.92202 

9.92534 

9.92859 

9.93175 

9.93484 

9.93785 

9.94079 

9.94365 

44 

.92208 

.92540 

.92864 

.93181 

.93489 

.93790 

.94084 

.94369 

48 

.92213 

.92545 

.92869 

.93186 

.93494 

.93795 

.94088 

.94374 

52 

.92219 

.92551 

.92875 

.93191 

.93499 

.93800 

.94093 

.94379 

56 

.92225 

.92556 

.92880 

.93196 

.93504 

.93805 

.94098 

.94383 

s 

8h  49"^ 

8h  53"' 

Sh  5Jm 

Qh  pn 

Qh   g»l 

Qh  Qm 

Qh   13m 

Qh   J  7m 

0 

9.92230 

9.92562 

9.92885 

9.93201 

9.93509 

9.93810 

9.94103 

9.94388 

4 

.92236 

.92567 

.92891 

.93207 

.93515 

.93815 

.94108 

.94393 

8 

.92241 

.92573 

.92896 

.93212 

.93520 

.93820 

.94112 

.94398 

12 

.92247 

.92578 

.92901 

.93217 

.93525 

.93825 

.94117 

.94402 

16 

.92253 

.92584 

.92907 

.93222 

.93530 

.93830 

.94122 

.94407 

20 

9.92258 

9.92589 

9.92912 

9.93227 

9.93535 

9.93835 

9.94127 

9.94412 

24 

.92264 

.92594 

.92917 

.93232 

.93540 

.93840 

.94132 

.94416 

28 

.92269 

.92600 

.92923 

.93238 

.93545 

.93845 

.94137 

.94421 

32 

.92275 

.92605 

.92928 

.93243 

.93550 

.93849 

.94141 

.94426 

36 

.92280 

.92611 

.92933 

.93248 

.93555 

.93854 

.94146 

.94430 

40 

9.92286 

9.92616 

9.92939 

9.93253 

9.93560 

9.93859 

9.94151 

9.94435 

44 

.92292 

.92622 

.92944 

.93258 

.93565 

.93864 

.94156 

.94440 

48 

.92297 

.92627 

.92949 

.93264 

.93570 

.93869 

.94161 

.94444 

52 

.92303 

.92633 

.92955 

.93269 

.93575 

.93874 

.94165 

.94449 

66 

.92308 

.92638 

.92960 

.93274 

.93580 

.93879 

.94170 

.94454 

s 

8f'  50"' 

8h  54'" 

8^  58"" 

gh  2"' 

Qh  Qm 

Qh  lo"' 

Qh  IJ^m 

Qh  IS"' 

0 

9.92314 

9.92643 

9.92965 

9.93279 

9.93585 

9.93884 

9.94175 

9.94458 

4 

.92319 

.92649 

.92970 

.93284 

.93590 

.93889 

.94180 

.94463 

8 

.92325 

.92654 

.92975 

.93289 

.93595 

.93894 

.94184 

.94468 

12 

.92330 

.92660 

.92981 

.93295 

.93600 

.93899 

.94189 

.94472 

16 

.92336 

.92665 

.92986 

.93300 

.93605 

.93904 

.94194 

.94477 

20 

9.92342 

9.92670 

9.92992 

9.93305 

9.93611 

9.93908 

9.94199 

9.94482 

24 

.92347 

.92676 

.92997 

.93310 

.93616 

.93913 

.94204 

.94486 

28 

.92353 

.92681 

.93002 

.93315 

.93621 

.93918 

.94208 

.94491 

32 

.92358 

.92687 

.93007 

.93320 

.93626 

.93923 

.94213 

.94496 

36 

.92364 

.92692 

.93013 

.93326 

.93631 

.93928 

.94218 

.94500 

40 

9.92369 

9.92698 

9.93018 

9.93331 

9.93636 

9.93933 

9.94223 

9.94505 

44 

.92375 

.92703 

.93023 

.93336 

.93641 

.93938 

.94227 

.94509 

48 

.92380 

.92708 

.93029 

.93341 

.93646 

.93943 

.94232 

.94514 

52 

.92386 

.92714 

.93D34 

.93346 

.93651 

.93948 

.94237 

.94519 

56 

.92391 

.92719 

.93039 

.93351 

.93656 

.93952 

.94242 

.94523 

s 

Sh  gpn 

8h  55"' 

8h  59'" 

Qh  gm 

Qh  jm 

Qh  ipn 

Qh  Igm. 

Qh  29'" 

0 

9.92397 

9.92725 

9.93044 

9.93356 

9.93661 

9.93957 

9.94246 

9.94528 

4 

.92402 

.92730 

.93050 

.93362 

.93666 

.93962 

.94251 

.94533 

8 

.92408 

.92735 

.93055 

.93367 

.93671 

.93967 

.94256 

.94537 

12 

.92413 

.92741 

.93060 

.93372 

.93676 

.93972 

.94261 

.94542 

16 

.92419 

.92746 

.93065 

.93377 

.93681 

.93977 

.94265 

.94546 

20 

9.92425 

9.92751 

9.93071 

9.93382 

9.93686 

9.93982 

9.94270 

9.94551 

24 

.92430 

.92757 

.93076 

.93387 

.93691 

.93987 

.94275 

.94556 

28 

.92436 

.92762 

.93081 

.93392 

.93696 

.93991 

.94280 

.94560 

32 

.92441 

.92768 

.93086 

.93397 

.93701 

.93996 

.94284 

.94565 

36 

.92447 

.92773 

.93092 

.93403 

.93706 

.94001 

.94289 

.94570 

40 

9.92452 

9.92778 

9.93097 

9.93408 

9.93711 

9.94006 

9.94294 

9.94574 

44 

.92458 

.92784 

.93102 

.93413 

.93716 

.94011 

.94299 

.94579 

48 

.92463 

.92789 

.93107 

.93418 

.93721 

.94016 

.94303 

.94583 

52 

.92469 

.92794 

.93113 

.93423 

.93726 

.94021 

.94308 

.94588 

56 

.92474 

.92800 

.93118 

.93428 

.93731 

.94026 

.94313 

.94593 

60 

9.92480 

9.92805 

9.93123 

9.93433 

9.93736 

9.94030 

9.94318 

9.94597 

Table  10.    Hayersine  Table 


279 


s 

gh^O'^ 

9h  24"" 

gh  28^ 

gh  5^m 

gh  S6"^ 

9h  4O"' 

9h  44"' 

9h  48"^ 

Hav. 

Hav. 

Hav. 

Hav. 

Hav. 

Hav. 

Hav. 

Hav. 

0 

9.94597 

9.94869 

9.95134 

9.95391 

9.95641 

9.95884 

9.96119 

9.96347 

4 

.94602 

.94874 

.95138 

.95396 

.95645 

.95888 

.96123 

.96351 

8 

.94606 

.94878 

.95143 

.95400 

.95049 

.95892 

.96127 

.96355 

12 

.94611 

.94883 

.95147 

.95404 

.95054 

.95896 

.96131 

.96359 

16 

.94616 

.94887 

.95151 

.95408 

.95658 

.95900 

.96135 

.96362 

20 

9.94620 

9.94892 

9.95156 

9.95412 

9.95662 

9.95904 

9.96139 

9.96366 

24 

.94625 

.94896 

.95160 

.95417 

.95666 

.95908 

.96142 

.96370 

28 

.94629 

.94901 

.95164 

.95421 

.95670 

.95912 

.96146 

.96374 

32 

.94634 

.94905 

.95169 

.95425 

.95674 

.95916 

.96150 

.96377 

36 

.94638 

.94909 

.95173 

.95429 

.95678 

.95920 

.96154 

.96381 

40 

9.94643 

9.94914 

9.95177 

9.95433 

9.95682 

9.95924 

9.96158 

9.96385 

44 

.94648 

.94918 

.95182 

.95438 

.95686 

.95928 

.96162 

.96388 

48 

.94652 

.94923 

.95186 

.95442 

.95690 

.95932 

.96165 

.96392 

52 

.94657 

.94927 

.95190 

.95440 

.95694 

.95936 

.96169 

.96396 

56 

.94661 

.94932 

.95195 

.95450 

.95699 

.95939 

.96173 

.96400 

s 

gh  21"^ 

gh25"' 

gh  29'^ 

gh  S3"' 

gh  37m 

gh  ^pn 

9h  45"^ 

9^'  49"^ 

0 

9.94666 

9.94936 

9.95199 

9.95454 

9.95703 

9.95943 

9.96177 

9.96403 

4 

.94670 

.94941 

.95203 

.95459 

.95707 

.95947 

.96181 

.96407 

8 

.94675 

.94945 

.95208 

.95463 

.95711 

.95951 

.96185 

.96411 

12 

.94680 

.94950 

.95212 

.95467 

.95715 

.95955 

.96188 

.96412 

16 

.94684 

.94954 

.95216 

.95471 

.95719 

.95959 

.96192 

.96418 

20 

9.94689 

9.94958 

9.95221 

9.95475 

9.95723 

9.95963 

9.96196 

9.96422 

24 

.94693 

.94963 

.95225 

.95480 

.95727 

.95967 

.96200 

.96426 

28 

.94698 

.94967 

.95229 

.95484 

.95731 

.95971 

.96204 

.96429 

32 

.94702 

.94972 

.95234 

.95488 

.95735 

.95975 

.96208 

.96433 

36 

.94707 

.94976 

.95238 

.95492 

.95739 

.95979 

.96211 

.96437 

40 

9.94711 

9.94981 

9.95242 

9.95496 

9.95743 

9.95983 

9.96215 

9.96440 

44 

.94716 

.94985 

.95246 

.95501 

.95747 

.95987 

.96219 

.96444 

48 

.94721 

.94989 

.95251 

.95505 

.95751 

.95991 

.96223 

.96448 

52 

.94725 

.94994 

.95255 

.95509 

.95755 

.95995 

.96227 

.96451 

56 

.94730 

.94998 

.95259 

.95513 

.95759 

.95999 

.96230 

.96455 

s 

gh  22^ 

gh  26^ 

gh  SO"" 

9'^  34"' 

gh  ss"' 

9^^  42"" 

9h  4&'' 

9h  50"^ 

0 

9.94734 

9.95003 

9.95264 

9.95517 

9.95763 

9.96002 

9.96234 

9.96459 

4 

.94739 

.95007 

.95268 

.95521 

.95768 

.96006 

.96238 

.96462 

8 

.94743 

.95011 

.95272 

.95526 

.95772 

.96010 

.96242 

.96466 

12 

.94748 

.95016 

.95276 

.95530 

.95776 

.96014 

.96246 

.96470 

16 

.94752 

.95020 

.95281 

.95534 

.95780 

.96018 

.96249 

.96473 

20 

9.94757 

9.95025 

9.95285 

9.95538 

9.95784 

9.96022 

9.96253 

9.96477 

24 

.94761 

.95029 

.95289 

.95542 

.95788 

.96026 

.96257 

.96481 

28 

.94766 

.95033 

.95294 

.95546 

.95792 

.96030 

.96261 

.96484 

32 

.94770 

.95038 

.95298 

.95550 

.95796 

.96034 

.96265 

.96488 

36 

.94774 

.95042 

.95302 

.95555 

.95800 

.96038 

.96268 

.96492 

40 

9.94779 

9.95047 

9.95306 

9.95559 

9.95804 

9.96042 

9.96272 

9.96495 

44 

.94784 

.95051 

.95311 

.95563 

.95808 

.96046 

.96276 

.96499 

48 

.94788 

.95055 

.95315 

.95567 

.95812 

.96049 

.96280 

.96503 

52 

.94793 

.95060 

.95319 

.95571 

.95816 

.96053 

.96283 

.96506 

56 

.94797 

.95064 

.95323 

.95575 

.95820 

.96057 

.96287 

.96510 

s 

gh  23^ 

gh27^ 

gh  SI"' 

9h  35'" 

gh  sg"' 

9h  43"' 

gh  J^jm 

gh  Jpn 

0 

9.94802 

9.95069 

9.95328 

9.95579 

9.95824 

9.96061 

9.96291 

9.96514 

4 

.94806 

.95073 

.95332 

.95584 

.95828 

.96065 

.96295 

.96517 

8 

.94811 

.95077 

.95336 

.95588 

.95832 

.96069 

.96299 

.96521 

12 

.94815 

.95082 

.95340 

.95592 

.95836 

.96073 

.96302 

.96525 

16 

.94820 

.95086 

.95345 

.95596 

.95840 

.96077 

.96306 

.96528 

20 

9.94824 

9.95090 

9.95349 

9.95600 

9.95844 

9.96081 

9.96310 

9.96532 

24 

.94829 

.95095 

.95353 

.95604 

.95848 

.96084 

.96314 

.96536 

28 

.94833 

.95099 

.95357 

.95608 

.95852 

.96088 

.96317 

.96539 

32 

.94838 

.95104 

.95362 

.95613 

.95856 

.96092 

.96321 

.96543 

36 

.94842 

.95108 

.95366 

.95617 

.95860 

.96096 

.96325 

.96547 

40 

9.94847 

9.95112 

9.95370 

9.95621 

9.95864 

9.96100 

9.96329 

9.96550 

44 

.94851 

.95117 

.95374 

.95625 

.95868 

.96104 

.96332 

.96554 

4S 

.94856 

.95121 

.95379 

.95629 

.95872 

.96108 

.96336 

.96557 

52 

.94860 

.95125 

.95383 

.95633 

.95876 

.96112 

.96340 

.96561 

56 

.94865 

.95130 

.95387 

.95637 

.95880 

.96115 

.96344 

.96565 

60 

9.94869 

9.95134 

9.95.391 

9.95641 

9.95884 

9.96119 

9.96347 

9.96568 

280 


Table  10.    Hayersine  Table 


s 

Qh  Q2"' 

9>'  66"' 

J  Oh  0"" 

lOh  4'« 

IQh  8'" 

IQh  12"" 

lOh  16'"  lOh  20"^ 

Hav. 

Hav. 

Hav. 

Hav. 

Hav. 

Hav. 

Hav. 

Hav. 

0 

9.96568 

9.96782 

9.96989 

9.97188 

9.97381 

9.9756.. 

9.97745 

9.97916 

4 

.96572 

.96786 

.96992 

.97192 

.97384 

.97569 

.97748 

.97919 

8 

.96576 

.96789 

.96996 

.97195 

.97387 

.97572 

.97751 

.97922 

12 

.96579 

.96793 

.96999 

.97198 

.97390 

.97575 

.97754 

.97925 

16 

.96583 

.96796 

.97002 

.97201 

.97393 

.97578 

.97756 

.97927 

20 

9.96586 

9.96800 

9.97006 

9.97205 

9.97397 

9.97581 

9.97759 

9.97930 

24 

.96590 

.96803 

.97009 

.97208 

.97400 

.97584 

.97762 

.97933 

28 

.96594 

.96807 

.97012 

.97211 

.97403 

.97587 

.97765 

.97936 

32 

.96597 

.96810 

.97016 

.97214 

.97406 

.97591 

.97768 

.97939 

36 

.96601 

.96814 

.97019 

.97218 

.97409 

.97594 

.97771 

.97941 

40 

9.96604 

9.96817 

9.97022 

9.97221 

9.97412 

9.97597 

9.97774 

9.97944 

44 

.96608 

.96821 

.97026 

.97224 

.97415 

.97600 

.97777 

.97947 

48 

.96612 

.96824 

.97029 

.97227 

.97418 

.97003 

.97780 

.97950 

62 

.96615 

.96827 

.97033 

.97231 

.97422 

.97600 

.97783 

.97953 

56 

.96619 

.96831 

.97036 

.97234 

.97425 

.97609 

.97785 

.97955 

s 

91^  63"' 

gh  gym 

2  Oh  1^ 

lOh  6"" 

IQh  9'" 

IQh  IS'n 

lOh  17'"  lOh  21"^ 

0 

9.96622 

9.96834 

9.97039 

9.97237 

9.97428 

9.97612 

9.97788 

9.97958 

4 

.96626 

.96837 

.97043 

.97240 

.97431 

.97615 

.97791 

.97961 

8 

.96630 

.96841 

.97046 

.97244 

.97434 

.97618 

.97794 

.97964 

12 

.96633 

.96845 

.97049 

.97247 

.97437 

.97621 

.97797 

.97966 

16 

.96637 

.96848 

.97052 

.97250 

.97440 

.97624 

.97800 

.97969 

20 

9.96640 

9.96852 

9.97056 

9.97253 

9.97443 

9.97627 

9.97803 

9.97972 

24 

.96644 

.96855 

.97059 

.97257 

.97447 

.97630 

.97806 

.97975 

28 

.96648 

.96859 

.97063 

.97260 

.97450 

.97633 

.97808 

.97977 

32 

.96651 

.96862 

.97066 

.97263 

.97453 

.97636 

.97811 

.97980 

36 

.96655 

.96866 

.97069 

.97266 

.97450 

.97639 

.97814 

.97983 

40 

9.96658 

9.96869 

9.97073 

9.97269 

9.97459 

9.97642 

9.97817 

9.97986 

44 

.96662 

.96873 

.97076 

.97273 

.97462 

.97645 

.97820 

.97988 

48 

.96665 

.96876 

.97079 

.97276 

.97465 

.97647 

.97823 

.97991 

62 

.96669 

.96879 

.97083 

.97279 

.97468 

.97650 

.97826 

.97994 

66 

.96673 

.96883 

.97086 

.97282 

.97471 

.97653 

.97829 

.97997 

s 

fjh  64"" 

gh  58"" 

lOh  2'" 

IQh  6'" 

I  Oh  10"' 

lOh  W" 

lOh  18'"  lOh  22'" 

0 

9.96676 

9.96886 

9.97089 

9.97285 

9.97474 

9.97656 

9.97831 

9.97999 

4 

.96680 

.96890 

.97093 

.97289 

.94478 

.97659 

.97834 

.98002 

8 

.96683 

.96894 

.97096 

.96292 

.97481 

.97662 

.97837 

.98005 

12 

.90687 

.96897 

.97099 

.97295 

.97484 

.97665 

.97840 

.98008 

16 

.96690 

.96900 

.97103 

.97298 

.97487 

.97668 

.97843 

.98010 

20 

9.96694 

9.96904 

9.97106 

9.97301 

9.97490 

9.97671 

9.97846 

9.98013 

24 

.96697 

.96907 

.97109 

.97305 

.97493 

.97674 

.97849 

.98016 

28 

.96701 

.96910 

.97113 

.97308 

.97496 

.97677 

.97851 

.98019 

32 

.96705 

.96914 

.97116 

.97311 

.97499 

.97680 

.97854 

.98021 

36 

.96708 

.96917 

.97119 

.97314 

.97502 

.97683 

.97857 

.98024 

40 

9.96712 

9.96921 

9.97123 

9.97317 

9.97505 

9.97686 

9.97860 

9.98027 

44 

.96715 

.96924 

.97126 

.97321 

.97508 

.97689 

.97863 

.98030 

48 

.96719 

.96928 

.97129 

.97324 

.97511 

.97692 

.97866 

.98032 

62 

.96722 

.96931 

.97132 

.97327 

.97514 

.97695 

.97868 

.98035 

66 

.96726 

.96934 

.97136 

.97330 

.97518 

.97698 

.97871 

.98038 

s 

gh   Jjm 

0^^  60"" 

lOh  3'" 

lOh  7'" 

IQh  11'" 

lOh  16'" 

IQh  19'"  lOh  23"' 

0 

9.96729 

9.96938 

9.97139 

9.97333 

9.97521 

9.97701 

9.97874 

9.98040 

4 

.96733 

.96941 

.97142 

.97337 

.97524 

.97704 

.97877 

.98043 

8 

.96736 

.96945 

.97146 

.97340 

.97527 

.97707 

.97880 

.98046 

12 

.96740 

.96948 

.97149 

.97343 

.97530 

.97710 

.97883 

.98049 

16 

.96743 

.96951 

.97152 

.97346 

.97533 

.97713 

.97885 

.98051 

20 

9.96747 

9.96955 

9.97156 

9.97349 

9.97536 

9.97716 

9.97888 

9.98054 

24 

.96750 

.96958 

.97159 

.97352 

.97539 

.97718 

.97891 

.98057 

28 

.96754 

.96962 

.97162 

.97350 

.97542 

.97721 

.97894 

.98059 

32 

.96758 

.96965 

.97165 

.97359 

.97545 

.97724 

.97897 

.98062 

36 

.98761 

.96968 

.97169 

.97362 

.97548 

.97727 

.97899 

.98065 

40 

9.96765 

9.96972 

9.97172 

9.97365 

9.97551 

9.97730 

9.97902 

9.98067 

44 

.96768 

.96975 

.97175 

.97368 

.97554 

.97733 

.97905 

.98070 

48 

.96772 

.96979 

.97179 

.97371 

.97557 

.97736 

.97908 

.98073 

62 

.96775 

.96982 

.97182 

.97375 

.97560 

.97739 

.97911 

.98076 

66 

.96779 

.96985 

.97185 

.97378 

.97563 

.97742 

.97914 

.98078 

60 

9.96782 

9.96989 

9.97188 

9.97381 

9.97566 

9.97745 

9.97916 

9.98081 

Table  10.    Haversiue  Table 


281 


s 

lOh  24'" 

10^  28'" 

lOh  32'" 

lOh  36'" 

lOt^  40^" 

lOh  44'" 

lOh  48^" 

lOh  52'" 

Hav. 

Hav. 

Hav. 

Hav. 

Hav. 

Hav. 

Hav. 

Hav. 

0 

9.98081 

9.98239 

9.98389 

9.98533 

9.98670 

9.98801 

9.98924 

9.99041 

A 

.98084 

.98241 

.98392 

.98536 

.98673 

.98803 

.98926 

.99043 

8 

.98086 

.98244 

.98394 

.98538 

.98675 

.98805 

.98928 

.99044 

12 

.98089 

.98246 

.98397 

.98540 

.98677 

.98807 

.98930 

.99046 

16 

.98092 

.98249 

.98399 

.98543 

.98679 

.98809 

\98932 

.99048 

20 

9.98094 

9.98251 

9.98402 

9.98545 

9.98681 

9.98811 

9.98934 

9.99050 

24- 

.98097 

.98254 

.98404 

.98547 

.98684 

.98813 

.98936 

.99052 

28 

.98100 

.98256 

.98406 

.98550 

.98686 

.98815 

.98938 

.99054 

32 

.98102 

.98259 

.98409 

.98552 

.98688 

.98817 

.98940 

.99056 

36 

.98105 

.98262 

.98411 

.98554 

.98690 

.98819 

.98942 

.99058 

40 

9.98108 

9.98264 

9.98414 

9.98557 

9.98692 

9.98822 

9.98944 

9.99059 

U 

.98110 

.98267 

.98416 

.98559 

.98695 

.98824 

.98946 

.99061 

48 

.98113 

.98269 

.98419 

.98561 

.98697 

.98826 

.98948 

.99063 

52 

.98116 

.98272 

.98421 

.98564 

.98699 

.98828 

.98950 

.99065 

66 

.98118 

.98274 

.98424 

.98566 

.98701 

.98830 

.98952 
10^  49'" 

.99067 
lOh  53'" 

s 

I  Oh  25'" 

IQh  29"^ 

lOh  33'" 

10h  37>n 

lOh  4V" 

W^  45'" 

0 

9.98121 

9.98277 

9.98426 

9.98568 

9.98703 

9.98832 

9.98954 

9.99069 

4 

.98124 

.98279 

.98428 

.98570 

.98706 

.98834 

.98956 

.99071 

8 

.98126 

.98282 

.98431 

.98573 

.98708 

.98836 

.98958 

.99072 

12 

.98129 

.98285 

.98433 

.98575 

.98710 

.98838 

.98960 

.99074 

16 

.98132 

.98287 

.98436 

.98577 

.•98712 

.98840 

.98962 

.99076 

20 

9.98134 

9.98290 

9.98438 

9.98580 

9.98714 

9.98842 

9.98964 

9.99078 

24 

.98137 

.98292 

.98440 

.98582 

.98717 

.98845 

.98966 

.99080 

28 

.98139 

.98295 

.98443 

.98584 

.98719 

.98847 

.98968 

.99082 

32 

.98142 

.98297 

.98445 

.98587 

.98721 

.98849 

.98970 

.99084 

36 

.98145 

.98300 

.98448 

.98589 

.98723 

.98851 

.98971 

.99085 

40 

9.98147 

9.98302 

9.98450 

9.98591 

9.98725 

9.98853 

9.98973 

9.99087 

44 

.98150 

.98305 

.98453 

.98593 

.98728 

.98855 

.98975 

.99089 

48 

.98153 

.98307 

.98455 

.98596 

.98730 

.98857 

.98977 

.99091 

52 

.98155 

.98310 

.98457 

.98598 

.98732 

.98859 

.98979 

.99093 

56 

.98158 

.98312 

.98460 

.98600 

.98734 

.98861 

.98981 
lOh  50'" 
9.98983 

.99095 
lOh  54'" 
9.99096 

s 

IQh  26'" 

lOf'  30'" 

lOh  34'" 

lOh  38'" 

IQh  42'" 

lO''  46'" 

0 

9.98161 

9.98315 

9.98462 

9.98603 

9.98736 

9.98863 

4 

.98163 

.98317 

.98465 

.98605 

.98738 

.98865 

.98985 

.99098 

8 

.98166 

.98320 

.98467 

.98607 

.98741 

.98867 

.98987 

.99100 

12 

.98168 

.98322 

.98469 

.98609 

.98743 

.98869 

.98989 

.99102 

16 

.98171' 

.98325 

.98472 

.98612 

.98745 

.98871 

.98991 

.99104 

20 

9.98174 

9.98327 

9.98474 

9.98614 

9.98747 

9.98873 

9.98993 

9.99106 

24 

.98176 

.98330 

.98476 

.98616 

.98749 

.98875 

.98995 

.99107 

28 

.98179 

.98332 

.98479 

.98619 

.98751 

.98877 

.98997 

.99109 

32 

.98182 

.98335 

.98481 

.98621 

.98754 

.98880 

.98999 

.99111 

36 

.98184 

.98337 

.98484 

.98623 

.98756 

.98882 

.99001 

.99113 

40 

9.98187 

9.98340 

9.98486 

9.98625 

9.98758 

9.98884 

9.99003 

9.99115 

44 

.98189 

.98342 

.98488 

.98628 

.98760 

.98886 

.99004 

.99116 

48 

.98192 

.98345 

.98491 

.98630 

.98762 

.98888 

.99006 

.99118 

52 

.98195 

.98347 

.98493 

.98632 

.98764 

.98890 

.99008 

.99120 

56 

.98197 

.98350 

.98496 

.98634 

.98766 

.98892 

.99010 

.99122 

s 

lOh  21'" 

IQh  31"^ 

lOh  35'" 

10'^  39'" 

IQh  43'" 

lOh  47'" 

IQh  51'" 

10^  55'" 

0 

9.98200 

9.98352 

9.98498 

9.98637 

9.98769 

9.98894 

9.99012 

9.99124 

4 

.98202 

.98355 

.98500 

.98639 

.98771 

.98896 

.99014 

.99126 

8 

.98205 

.98357 

.98503 

.98641 

.98773 

.98898 

.99016 

.99127 

12 

.98208 

.98360 

.98505 

.98643 

.98775 

.98900 

.99018 

.99129 

16 

.98210 

.98362 

.98507 

.98646 

.98777 

.98902 

.99020 

.99131 

20 

9.98213 

9.98365 

9.98510 

9.98648 

9.98779 

9.98904 

9.99022 

9.99133 

24 

.98215 

.98367 

.98512 

.98650 

.98781 

.98906 

.99024 

.99135 

28 

.98218 

.98370 

.98514 

.98652 

.98784 

.98908 

.99026 

.99136 

32 

.98221 

.98372 

.98517 

.98655 

.98786 

.98910 

.99027 

.99138 

36 

.98223 

.98375 

.98519 

.98657 

.98788 

.98912 

.99029 

.99140 

40 

9.98226 

9.98377 

9.98521 

9.98659 

9.98790 

9.98914 

9.99031 

9.99142 

44 

.98228 

.98379 

.98524 

.98661 

.98792 

.98916 

.99033 

.99143 

48 

.98231 

.98382 

.98526 

.98664 

.98794 

.98918 

.99035 

.99145 

52 

.98233 

.98384 

.98529 

.98666 

.98796 

.98920 

.99037 

.99147 

56 

.98236 

.98387 

.98531 

.98668 

.98798 

.98922 

.99039 

.99149 

60 

9.98239 

9.98389 

9.98533 

9.98670 

9.98801 

9.98924 

9.99041 

9.99151 

282 


Table  10.    Haversine  Table 


s 

IQh  56'" 

llh  0"' 

llh  J^m 

llh  8'^ 

llh  12'" 

llh  16"^ 

llh  20'" 

llh  24"^ 

Hav. 

Hav. 

Hav. 

Hav. 

Hav. 

Hav. 

Hav. 

Hav. 

0 

9.99151 

9.99254 

9.99350 

9.99440 

9.99523 

9.99599 

9.99669 

9.99732 

4 

.99152 

.99255 

.99352 

.99441 

.99524 

.99600 

.99670 

.99733 

8 

.99154 

.99257 

.99353 

.99443 

.99526 

.99602 

.99671 

.99734 

12 

.99156 

.99259 

.99355 

.99444 

.99527 

.99603 

.99672 

.99735 

16 

.99158 

.99260 

.99356 

.99446 

.99528 

.99604 

.99673 

.99736 

20 

9.99159 

9.99262 

9.99358 

9.99447 

9.99529 

9.99605 

9.99674 

9.99737 

24 

.99161 

.99264 

.99359 

.99448 

.99531 

.99606 

.99675 

.'99738 

28 

.99163 

.99265 

.99361 

.99450 

.99532 

.99608 

.99677 

.99739 

32 

.99165 

.99267 

.99362 

.99451 

.99533 

.99609 

.99678 

.99740 

36 

.99166 

.99269 

.99364 

.99453 

.99535 

.99610 

.99679 

.99741 

40 

9.99168 

9.99270 

9.99366 

9.99454 

9.99536 

9.99611 

9.99680 

9.99742 

44 

.99170 

.99272 

.99367 

.99456 

.99537 

.99612 

.99681 

.99743 

48 

.99172 

.99274 

.99369 

.99457 

.99539 

.99614 

.99682 

.99744 

52 

.99173 

.99275 

.99370 

.99458 

.99540 

.99615 

.99683 

.99745 

56 

.99175 

.99277 

.99372 

.99460 

.99541 

.99616 

.99684 

.99746 

s 

IQh  5T^ 

llh  im 

llh  5m 

llh  gm 

llh  ism 

llh  ijm 

llh  21^ 

llh  25"" 

0 

9.99177 

9.99278 

9.99373 

9.99461 

9.99543 

9.99617 

9.99685 

9.99747 

4 

.99179 

.99280 

.99375 

.99463 

.99544 

.99618 

.99686 

.99748 

8 

.99180 

.99282 

.99376 

.99464 

.99545 

.99620 

.99687 

.99748 

12 

.99182 

.99283 

.99378 

.99465 

.99546 

.99621 

.99688 

.99749 

16 

.99184 

.99285 

.99379 

.99467 

.99548 

.99622 

.99690 

.99750 

20 

9.99186 

9.99287 

9.99381 

9.99468 

9.99549 

9.99623 

9.99691 

9.99751 

24 

.99187 

.99288 

.99382 

.99470 

.99550 

.99624 

.99692 

.99752 

28 

.99189 

.99290 

.99384 

.99471 

.99552 

.99626 

.99693 

.99753 

32 

.99191 

.99291 

.99385 

.99472 

.99553 

.99627 

.99694 

.99754 

36 

.99193 

.99293 

.99387 

.99474 

.99554 

.99628 

.99695 

.99755 

40 

9.99194 

9.99295 

9.99388 

9.99475 

9.99555 

9.99629 

9.99696 

9.99756 

44 

.99196 

.99296 

.99390 

.99477 

.99557 

.99630 

.99697 

.99757 

48 

.99198 

.99298 

.99391 

.99478 

.99558 

.99631 

.99698 

.99758 

52 

.99200 

.99300 

.99393 

.99479 

.99559 

.99633 

.99699 

.99759 

56 

.99201 

.99301 

.99394 

.99481 

.99561 

.99634 

.99700 

.99760 

s 

10^  58"" 

llh  2'"- 

llh  em 

llh  lO'n 

llh  IJ^'n 

llh  18"^ 

llh  22"' 

llh  26'" 

0 

9.99203 

9.99303 

9.99396 

9.99482 

9.99562 

9.99635 

9.99701 

9.99761 

4 

.99205 

.99304 

.99397 

.99484 

.99563 

.99636 

.99702 

.99762 

8 

.99206 

.99306 

.99399 

.99485 

.99564 

.99637 

.99703 

.99763 

12 

.99208 

.99308 

.99400 

.99486 

.99566 

.99638 

.99704 

.99764 

16 

.99210 

.99309 

.99402 

.99488 

.99567 

.99639 

.99705 

.99765 

20 

9.99212 

9.99311 

9.99403 

9.99489 

9.99568 

9.99641 

9.99706 

9.99766 

24 

.99213 

.99312 

.99405 

.99490 

.99569 

.99642 

.99707 

.99766 

28 

.99215 

.99314 

.99406 

.99492 

.99571 

.99643 

.99708 

.99767 

32 

.99217 

.99316 

.99408 

.99493 

.99572 

.99644 

.99710 

.99768 

36 

.99218 

.99317 

.99409 

.99495 

.99573 

.99645 

.99711 

.99769 

40 

9.99220 

9.99319 

9.99411 

9.99496 

9.99575 

9.99646 

9.99712 

9.99770 

44 

.99222 

.99320 

.99412 

.99497 

.99576 

.99648 

.99713 

.99771 

48 

.99223 

.99322 

.9.9414 

.99499 

.99577 

.99649 

.99714 

.99772 

52 

.99225 

.99324 

.99415 

.99500 

.99578 

.99650 

.99715 

.99773 

56 

.99227 

.99325 

.99417 

.99501 

.99580 

.99651 

.99716 

.99774 

s 

IQh  59"" 

llh  3m 

llh  7m 

llh  11"^ 

llh  ism 

llh  igm 

llh  23'" 

llh  27'" 

0 

9.99229 

9.99327 

9.99418 

9.99503 

9.99581 

9.99652 

9.99717 

9.99774 

4 

.99230 

.99328 

.99420 

.99504 

.99582 

.99653 

.99718 

.99775 

8 

.99232 

.99330 

.99421 

.99505 

.99583 

.99654 

.99719 

.99776 

12 

.99234 

.99331 

.99422 

.99507 

.99584 

.99655 

.99720 

.99777 

16 

.99235 

.99333 

.99424 

.99508 

.99586 

.99657 

.99721 

.99778 

20 

9.99237 

9.99335 

9.99425 

9.99510 

9.99587 

9.99658 

9.99722 

9.99779 

24 

.99239 

.99336 

.99427 

.99511 

.99588 

.99659 

.99723 

.99780 

28 

.99240 

.99338 

.99429 

.99512 

.99589 

.99660 

.99724 

.99781 

32 

.99242 

.99339 

.99430 

.99514 

.99591 

.99661 

.99725 

.99782 

36 

.99244 

.99341 

.99431 

.99515 

.99592 

.99662 

.99726 

.99783 

40 

9.99245 

9.99342 

9.99433 

9.99516 

9.99593 

9.99663 

9.99727 

9.99784 

44 

.99247 

.99344 

.99434 

.99518 

.99594 

.99664 

.99728 

.99785 

48 

.99249 

.99345 

.99436 

.99519 

.99596 

.99666 

.99729 

.99786 

52 

.99250 

.99347 

.99437 

.99520 

.99597 

.99667 

.99730 

.99786 

56 

.99252 

.99349 

.99438 

.99522 

.99598 

.99668 

.99731 

.99787 

60 

9.99254 

9.99350 

9.99440 

9.99523 

9.99599 

9.99669 

9.99732 

9.99788 

Table  10.    Haversine  Table 


283 


s 

llh  28"' 

llh  32'n 

llh  36"' 

llh  ^0"' 

llh  J^jm 

llh  ^gm 

llh  g£m 

llh  56"" 

Hav. 

Hav. 

Hav. 

Hav. 

9.99917 

Hav. 

Hav. 

Hav. 

Hav. 

0 

9.99788 

9.99838 

9.99881 

9.99947 

9.99970 

9.99987 

9.99997 

A 

.99789 

.99839 

.99882 

.99918 

.99948 

.99971 

.99987 

.99997 

8 

.99790 

.99839 

.99882 

.99918 

.  .99948 

.99971 

.99987 

.99997 

12 

.99791 

.99840 

.99883 

.99919 

.99948 

.99971 

.99987 

.99997 

16 

.99792 

.99841 

.99884 

.99919 

.99949 

.99972 

.99988 

.99997 

20 

9.99793 

9.99842 

9.99884 

9.99920 

9.99949 

9.99972 

9.99988 

9.99997 

24 

.99793 

.99842 

.99885 

.99921 

.99950 

.99972 

.99988 

.99997 

28 

.99794 

.99843 

.99885 

.99921 

.99950 

.99973 

.99988 

.99997 

32 

.99795 

.99844 

.99886 

.99922 

.99951 

.99973 

.99988 

.99998 

36 

.99796 

.99845 

.99887 

.99922 

.99951 

.99973 

.99989 

.99998 

40 

9.99797 

9.99845 

9.99887 

9.99923 

9.99951 

9.99973 

9.99989 

9.99998 

44 

.99798 

.99846 

.99888 

.99923 

.99952 

.99974 

.99989 

.99998 

48 

.99799 

.99847 

.99889 

.99924 

.99952 

.99974 

.99989 

.99998 

52 

.99800 

.99848 

.99889 

.99924 

.99953 

.99974 

.99989 

.99998 

56 

.99800 

.99848 

.99890. 

.99925 

.99953 

.99975 

.99990 

.99998 

s 

llh  29"' 

llh  33>n 

llh  37>n 

llh  J^pn 

llh  l^m 

llh  J^fjm 

llh  53'" 

llh  57m 

0 

9.99801 

9.99849 

9.99891 

9.99925 

9.99953 

9.99975 

9.99990 

9.99998 

4 

.99802 

.99850 

.99891 

.99926 

.99954 

.99975 

.99990 

.99998 

8 

.99803 

.99851 

.99892 

.99926 

.99954 

.99976 

.99990 

.99998 

12 

.99804 

.99851 

.99893 

.99927 

.99954 

.99976 

.99990 

.99998 

16 

.99805 

.99852 

.99893 

.99927 

.99955 

.99976 

.99991 

.99998 

20 

9.99805 

9.99853 

9.99894 

9.99928 

9.99955 

9.99976 

9.99991 

9.99999 

24 

.99806 

.99854 

.99894 

.99928 

.99956 

.99977 

.99991 

.99999 

28 

.99807 

.99854 

.99895 

.99929 

.99956 

.99977 

.99991 

.99999 

32 

.99808 

.99855 

.99896 

.99929 

.99957 

.99977 

.99991 

.99999 

36 

.99809 

.99856 

.99896 

.99930 

.99957 

.99978 

.99992 

.99999 

40 

9.99810 

9.99857 

9.99897 

9.99931 

9.99958 

9.99978 

9.99992 

9.99999 

44 

.99811 

.99857 

.99897 

.99931 

.99958 

.99978 

.99992 

.99999 

48 

.99811 

.99858 

.99898 

.99932 

.99958 

.99978 

.99992 

.99999 

52 

.99812 

.99859 

.99899 

.99932 

.99959 

.99979 

.99992 

.99999 

56 

.99813 

.99859 

.99899 

.99933 

.99959 

.99979 

.99992 

.99999 

s 

llh  30"' 

llh  3nn 

llh  3sm 

llh  42"' 

llh  ^0m 

llh  50"' 

llh    -j^m 

llh  gs"' 

0 

9.99814 

9.99860 

9.99900 

9.99933 

9.99959 

9.99979 

9.99993 

9.99999 

4 

.99815 

.99861 

.99901 

.99934 

.99960 

.99980 

.99993 

.99999 

8 

.99815 

.99862 

.99901 

.99934 

.99960 

.99980 

.99993 

.99999 

12 

.99816 

.99862 

.99902 

.99935 

.99961 

.99980 

.99993 

.99999 

16 

.99817 

.99863 

.99902 

.99935 

.99961 

.99980 

.99993 

.99999 

20 

9.99818 

9.99864 

9.99903 

9.99935 

9.99961 

9.99981 

9.99993 

9.99999 

24 

.99819 

.99864 

.99904 

.99936 

.99962 

.99981 

.99994 

.99999 

28 

.99820 

.99865 

.99904 

.99936 

.99962 

.99981 

.99994 

.00000 

32 

.99820 

.99866 

.99905 

.99937 

.99963 

.99981 

.99994 

.00000 

36 

.99821 

.99867 

.99905 

.99937 

.99963 

.99982 

.99994 

.00000 

40 

9.99822 

9.99867 

9.99906 

9.99938 

9.99963 

9.99982 

9.99994 

0.00000 

44 

.99823 

.99868 

.99906 

.99938 

.99964 

.99982 

.99994 

.00000 

48 

.99824 

.99869 

.99907 

.99939 

.99964 

.99983 

.99994 

.00000 

52 

.99824 

.99869 

.99908 

.99939 

.99964 

.99983 

.99995 

.00000 

56 

.99825 

.99870 

.99908 

.99940 

.99965 

.99983 

.99995 

.00000 

s 

llh  31'n 

llh  35"' 

llh  39"' 

llh  J^3m 

llh  ^jm 

llh  si>n 

llh  o5"' 

llh  59"' 

0 

9.99826 

9.99871 

9.99909 

9.99940 

9.99965 

9.99983 

9.99995 

0.00000 

4 

.99827 

.99871 

.99909 

.99941 

.99965 

.99983 

.99995 

.00000 

8 

.99828 

.99872 

.99910 

.99941 

.99966 

.99984 

.99995 

.00000 

12 

.99828 

.99873 

.99911 

.99942 

.99966 

.99984 

.99995 

.00000 

16 

.99829 

.99874 

.99911 

.99942 

.99966 

.99984 

.99995 

.00000 

20 

9.99830 

9.99874 

9.99912 

9.99943 

9.99967 

9.99984 

9.99996 

0.00000 

24 

.99831 

.99875 

.99912 

.99943 

.99967 

.99985 

.99996 

.00000 

28 

.99832 

.99876 

.99913 

.99943 

.99968 

.99985 

.99996 

.00000 

32 

.99832 

.99876 

.99913 

.99944 

.99968 

.99985 

.99996 

.00000 

36 

.99833 

.99877 

.99914 

.99944 

.99968 

.99985 

.99996 

.00000 

40 

9.99834 

9.99878 

9.99915 

9.99945 

9.99969 

9.99986 

9.99996 

0.00000 

44 

.99835 

.99878 

.99915 

.99945 

.99969 

.99986 

.99996 

.00000 

48 

.99836 

.99879 

.99916 

.99946 

.99969 

.99986 

.99996 

.00000 

52 

.99836 

.99880 

.99916 

.99946 

.99970 

.99986 

.99996 

.00000 

56 

.99837 

.99880 

.99917 

.99947 

.99970 

.99987 

.99997 

.00000 

60 

9.99838 

9.99881 

9.99917 

9.99947 

9.99970 

9.99987 

9.99997 

0.00000 

284 


Table  11.    Azimuth 


T, 

The  Ship's  Apparent  Time  for  a 

Sun  Observation, 

or  thi 

3  Hour-angle  fo 

R  A  StAI 

I  Observation 

Use 

Use 

These 

i2h    gm 

12h    8"' 

I2h  IB"' 

12h  24"^ 

12h  32"" 

12^  40^ 

12h  48"" 

12^56^ 

These 

IN 

^ 

■< —   in 

Fore- 

24    0 

23  52 

23  U 

23  36 

23  28 

23  20 

23  12 

23    4 

Fore- 

noon 

noon 

Use 

Use 

These 

Qh     Om 

Qh    s^i 

0^16"" 

Qh  24^ 

0^32'" 

0^40"" 

0^48"^ 

OhQ&n 

These 

IN 

^ 

<—    IN 

After- 

12       0 

11    52 

11  44 

11    36 

11    28 

11   20 

11    12 

11     4 

After- 

noon 

noon 

0° 

0 

349 

698 

1045 

1392 

1737 

2079 

2419 

0" 

2 

0 

349 

697 

1045 

1391 

1736 

2078 

2417 

2 

4 

0 

348 

696 

1042 

1389 

1732 

2074 

2413 

4 

6 

0 

347 

694 

1040 

1384 

1726 

2067 

2406 

6 

8 

0 

346 

691 

1035 

1378 

1720 

2059 

2395 

8 

10 

0 

344 

687 

1029 

1371 

1710 

2047 

2382 

10 

12 

0 

341 

682 

1022 

1361 

1698 

2033 

2367 

12 

14 

0 

339 

677 

1015 

1351 

1685 

2018 

2347 

14 

16 

0 

336 

671 

1005 

1338 

1669 

1998 

2326 

16 

18 

0 

332 

663 

994 

1323 

1651 

1977 

2301 

18 

20 

0 

328 

656 

982 

1308 

1632 

1954 

2274 

20 

22 

0 

324 

647 

969 

1290 

1610 

1928 

2244 

22 

24 

0 

319 

637 

955 

1272 

1586 

1900 

2210 

24 

26 

0 

314 

627 

940 

1251 

1561 

1868 

2174 

26 

28 

0 

308 

616 

923 

1228 

1533 

1835 

2136 

28 

30 

0 

302 

604 

905 

1205 

1504 

1801 

2095 

30 

32 

0 

296 

592 

886 

1180 

1472 

1763 

2051 

32 

34 

0 

289 

578 

867 

1153 

1440 

1724 

2005 

34 

36 

0 

282 

564 

846 

1126 

1405 

1682 

1957 

36 

m 

38 

0 

275 

550 

824 

1096 

1369 

1639 

1906 

38 

o 

40 

0 

267 

534 

801 

1066 

1330 

1592 

1853 

40 

M 

H 

H 

42 

0 

259 

518 

777 

1034 

1290 

1545 

1798 

42 

CI 
►3 

•^ 
Z 

44 

0 

251 

502 

752 

1001 

1249 

1496 

1740 

44 

H 

46 

0 

242 

485 

726 

967 

1206 

1444 

1681 

46 

48 

0 

234 

467 

699 

931 

1162 

1391 

1619 

48 

< 

Q 

50 

0 

224 

448 

672 

895 

1116 

1337 

1555 

50 

52 

0 

215 

429 

644 

857 

1069 

1280 

1489 

52 

54 

0 

205 

411 

615 

818 

1021 

1222 

1422 

54 

56 

0 

195 

390 

585 

778 

971 

1162 

1353 

56 

58 

0 

185 

370 

554 

738 

920 

1102 

1282 

58 

60 

0 

175 

349 

523 

696 

868 

1040 

1210 

60 

62 

0 

164 

328 

490 

653 

815 

976 

1136 

62 

64 

0 

153 

306 

458 

610 

761 

911 

1060 

64 

66 

0 

142 

284 

425 

566 

706 

846 

984 

66 

68 

0 

131 

201 

392 

521 

651 

779 

906 

68 

70 

0 

119 

239 

358 

476 

594 

711 

827 

70 

72 

0 

108 

216 

323 

430 

537 

643 

748 

72 

74 

0 

96 

192 

288 

384 

479 

573 

667 

74 

76 

0 

84 

169 

253 

337 

420 

503 

585 

76 

78 

0 

73 

145 

217 

289 

361 

432 

503 

78 

80 

0 

61 

121 

182 

242 

302 

361 

420 

80 

82 

0 

49 

97 

146 

194 

242 

289 

337 

82 

84 

0 

36 

73 

109 

146 

182 

217 

253 

84 

86 

0 

24 

49 

73 

97 

121 

145 

169 

86 

88 

0 

12 

24 

36 

49 

61 

73 

84 

88 

Use 

Use 

These 

0° 

2° 

4° 

6° 

8° 

10° 

12° 

14° 

These 

in 

— >• 

■< —   in 

Fore- 

180 

178 

176 

174 

172 

170 

168 

166 

Fore- 

noon 

noon 

Use 

Use 

These 

180° 

182° 

184° 

186° 

188° 

190° 

192° 

194° 

These 

IN 

> 

-< —   in 

After- 

360 

358 

356 

354 

352 

350 

348 

346 

After- 

noon 

True 

-  Beauin 

G  OR  Azi 

muth 

noon 

Table  11.    Azimuth 


285 


T,  The 

Ship's 

(Apparent  Time  for  a  Sun  Observation, 

Ol 

t  the  H( 

)UR-ANGL 

E  FOR  A 

Star  Ob 

servatic 

)N 

Use 

Use 

These 

ISh     jrn 

13h  12"" 

13h  20"" 

13^28^ 

IShSQm 

I3h  44^ 

13^  52"" 

These 

IN 

—> 

<. IN 

Fore- 

22 56 

22  48 

22  40 

22  32 

22   24 

22   16 

22     8 

Fore- 

noon 

noon 

Use 

Use 

These 

Ih     jm 

Ih  12"' 

Ih  20"^ 

ih  Ogm 

IhSQm 

1^44^ 

lh52r 

These 

IN 

— >- 

<. —    IN 

After- 

10  56 

10   48 

10  40 

10  32 

10  24 

10   16 

10     8 

After- 

noon 

noon 

0° 

2756 

3090 

3421 

3746 

4067 

4383 

4695 

C° 

2 

2755 

3088 

3418 

3744 

4065 

4381 

4692 

2 

4 

2750 

3082 

3412 

3737 

4058 

4373 

4684 

4 

6 

2742 

3073 

3402 

3726 

4044 

4360 

4669 

6 

8 

2730 

3060 

3387 

3710 

4028 

4341 

4649 

8 

10 

2714 

3043 

3368 

3689 

4005 

4317 

4624 

10 

12 

2696 

3023 

3346 

3664 

3978 

4287 

4592 

12 

14 

2674 

2998 

3319 

3635 

3947 

4253 

4555 

14 

16 

2650 

2970 

3288 

3600 

3910 

4214 

4513 

16 

18 

2622 

2939 

3253 

3563 

3868 

4170 

4465 

18 

20 

2590 

2903 

3214 

3521 

3821 

4119 

4412 

20 

22 

2556 

2865 

3171 

3473 

3771 

4064 

4353 

22 

24 

2519 

2823 

3125 

3422 

3715 

4005 

4288 

24 

26 

2477 

2777 

3074 

3367 

3656 

3941 

4220 

26 

28 

2434 

2729 

3020 

3308 

3591 

3871 

4145 

28 

30 

2387 

2676 

2962 

3244 

3522 

3796 

4065 

30 

32 

2337 

2620 

2900 

3177 

3449 

3718 

3981 

32 

34 

2285 

2563 

2835 

3106 

3372 

3634 

3892 

34 

36 

2230 

2500 

2767 

3031 

3291 

3546 

3798 

36 

m 

38 

2172 

2435 

2696 

2952 

3205 

3454 

3699 

38 

O 

40 

2112 

2367 

2620 

2869 

3116 

3358 

3596 

40 

aj 

P 

42 

2048 

2297 

2542 

2784 

3023 

3258 

3489 

42 

< 

44 

1983 

2223 

2460 

2695 

2925 

3154 

3377 

44 

3 

46 

1914 

2147 

2375 

2602 

2826 

3045 

3261 

46 

H 

48 

1845 

2067 

2289 

2507 

2721 

2934 

3142 

48 

< 

Q 

50 

1772 

1986 

2199 

2408 

2614 

2818 

3018 

50 

52 

1697 

1902 

2106 

2306 

2504 

2699 

2891 

52 

54 

1620 

1818 

2010 

2202 

2391 

2577 

2759 

54 

56 

1541 

1728 

1913 

2094 

2275 

2451 

2625 

56 

58 

1461 

1638 

1812 

1986 

2155 

2324 

2488 

58 

60 

1378 

1545 

1710 

1874 

2033 

2192 

2347 

60 

62 

1294 

1451 

1600 

1759 

1909 

2058 

2204 

62 

64 

1209 

1355 

1499 

1643 

1783 

1922 

2058 

64 

66 

1121 

1257 

1391 

1524 

1654 

1783 

1909 

66 

68 

1032 

1158 

1281 

1404 

1524 

1643 

1759 

68 

70 

943 

1057 

1169 

1281 

1391 

1499 

1606 

70 

72 

852 

955 

1057 

1158 

1257 

1355 

1451 

72 

74 

760 

852 

943 

1032 

1121 

1209 

1294 

74 

76 

667 

748 

827 

906 

984 

1060 

1136 

76 

78 

573 

643 

711 

779 

846 

911 

976 

78 

80 

479 

537 

594 

651 

706 

761 

815 

80 

82 

384 

430 

476 

521 

566 

610 

653 

82 

84 

288 

323 

358 

392 

425 

458 

491 

84 

86 

192 

216 

239 

261 

290 

306 

328 

86 

88 

96 

108 

119 

131 

142 

153 

164 

88 

Use 

Use 

These 

16° 

18° 

20° 

22° 

24° 

26° 

28° 

These 

IN 

— > 

•< —   IN 

Fore- 

164 

162 

160 

158 

156 

154 

152 

Fore- 

noon 

noon 

Use 

Use 

These 

196° 

198° 

200° 

202° 

204° 

206° 

208° 

These 

IN 

— >- 

-< —   IN 

After- 

344 

342 

340 

338 

336 

334 

332 

After- 

noon 

1 

'rue  Be 

\RING  OR 

AZIMUT 

H 

noon 

286 


Table  11.    Azimuth 


T, 

The  Ship's  Apparent  Time  for  a 

Sun  Observation.     1 

OR  THE 

HoUR-i* 

^NGLE  FO 

R  A  StAI 

I  Obser\ 

'^ATION 

Use 

Use 

These 

14h     0'" 

14h      §m 

14^  16"" 

14^24"^ 

14^32^ 

14^  40"^ 

14h  48"^ 

14^56^ 

These  | 

IN   — > 

^ 

IN 

Fore- 

23    0 

21    52 

21  44 

21    36 

21    28 

21    20 

21    12 

21     4 

Fore- 

noon 

noon 

Use 

Use 

These 

2h    0"" 

2h    8"" 

2hiem 

2h24^ 

2h32^ 

2^40"" 

2^  48^ 

2h  56"" 

These 

IN   — > 

^ 

IN 

After- 

10    0 

9  52 

9  44 

9  36 

9  28 

9  20 

9   12 

9     4 

After- 

noon 

noon 

0° 

5000 

5299 

5593 

5878 

6156 

6428 

6691 

6947 

0° 

2 

4997 

5297 

5589 

5875 

6153 

6424 

6688 

6942 

2 

4 

4987 

5286 

5578 

5864 

6142 

6412 

6676 

6929 

4 

6 

4973 

5270 

5562 

5845 

6124 

6393 

6655 

6908 

6 

8 

4951 

5248 

5538 

5821 

6096 

6365 

6627 

6879 

8 

10 

4923 

5219 

5507 

5789 

6063 

6330 

6591 

6841 

10 

12 

4891 

5183 

5470 

5749 

6022 

6288 

6545 

6795 

12 

14 

4852 

5141 

5426 

5703 

5973 

6237 

6492 

6741 

14 

16 

4806 

5094 

5375 

5650 

5919 

6179 

6433 

6677 

16 

18 

4755 

5040 

5319 

5590 

5856 

6113 

6363 

6607 

18 

20 

4699 

4979 

5255 

5524 

5785 

6040 

6288 

6528 

20 

22 

4635 

4914 

5184 

5450 

5709 

5960 

6204 

6440 

22 

24 

4567 

4841 

5109 

5370 

5624 

5872 

6112 

6346 

24 

26 

4493 

4763 

5025 

5283 

5534 

5777 

6015 

6243 

26 

28 

4415 

4678 

4938 

5190 

5437 

5675 

5907 

6134 

28 

30 

4330 

4588 

4843 

5091 

5332 

5567 

5794 

6016 

30 

32 

4240 

4493 

4742 

4984 

5222 

5451 

5674 

5891 

32 

34 

4145 

4393 

4635 

4873 

5104 

5328 

5547 

5758 

34 

36 

4044 

4287 

4524 

4755 

4980 

5200 

5414 

5619 

36 

CO 

38 

3941 

4176 

4407 

4633 

4852 

5065 

5272 

5474 

38 

o 

40 

3830 

4060 

4284 

4503 

4717 

4923 

5126 

5321 

40 

00 

H 

42 

3715 

3939 

4156 

4368 

4575 

4776 

4973 

5162 

42 

Q 

< 

44 

3596 

3812 

4023 

4229 

4429 

4624 

4812 

4997 

44 

3 

46 

3473 

3681 

3885 

4083 

4277 

4465 

4648 

4825 

46 

48 

3346 

3546 

3742 

3932 

4120 

4301 

4477 

4648 

48 

< 

Q 

50 

3214 

3406 

3594 

3779 

3958 

4131 

4301 

4465 

50 

52 

3078 

3263 

3443 

3619 

3790 

3958 

4120 

4277 

52 

54 

2939 

3115 

3287 

3454 

3619 

3779 

3932 

4083 

54 

56 

2796 

2963 

3127 

3287 

3443 

3594 

3742 

3885 

56 

58 

2650 

2808 

2963 

3115 

3263 

3406 

3546 

3681 

58 

60 

2500 

2650 

2796 

2939 

3078 

3214 

3346 

3473 

60 

62 

2347 

2488 

2625 

2760 

2891 

3018 

3142 

3261 

62 

64 

2192 

2324 

2451 

2577 

2699 

2818 

2934 

3045 

64 

66 

2033 

2155 

2275 

2391 

2504 

2614 

2721 

2826 

66 

68 

1874 

1986 

2094 

2202 

2306 

2408 

2507 

2602 

68 

70 

1710 

1812 

1913 

2010 

2106 

2199 

2289 

2375 

70 

72 

1545 

1638 

1728. 

1817 

1902 

1986 

2067 

2147 

72 

74 

1378 

1461 

1541 

1620 

1697 

1720 

1845 

1914 

74 

76 

1210 

1282 

1353 

1422 

1489 

1555 

1619 

1681 

76 

78 

1040 

1102 

1162 

1222 

1280 

1337 

1391 

1444 

78 

80 

868 

920 

971 

1021 

1069 

1116 

1162 

1206 

80 

82 

696 

738 

778 

818 

857 

895 

931 

967 

82 

84 

523 

554 

585 

615 

644 

672 

699 

726 

84 

86 

349 

370 

390 

411 

429 

448 

467 

485 

86 

88 

175 

185 

195 

205 

215 

224 

234 

242 

88 

Use 

Use 

These 

30° 

32° 

34° 

36° 

38° 

40° 

42° 

44° 

These 

IN   — ^ 

< 

IN 

Fore- 

150 

148 

146 

144 

142 

140 

138 

136 

Fore- 

noon 
Use 

noon 

Use 

These 

210° 

212° 

214° 

216° 

218° 

220° 

222° 

224° 

These 

IN   — >■ 

IN 

After- 

330 

328 

326 

324 

322 

320 

318 

316 

After- 

noon 

True 

)  Bearin 

G  OR  Azi 

muth 

noon 

Table  11.    Azimuth 


287 


T,  The  Ship's 

(Apparent  Time  for  a  Sun  Observation, 

, 

o 

R  THE  H( 

DUR-ANGLE  FOR  A 

Star  Observation 

Use 

Use 

These 

loh    4'" 

15h  12^ 

15f^  20"" 

15^  28"" 

15h  36"" 

15h  44^ 

15^  52"^ 

These 

IN   -^- 

■< IN 

Fore- 

20   56 

20   48 

20   40 

20   32 

20  24 

20   16 

20     8 

Fore- 

NOON 

Use 

noon 

Use 

These 

3h    4'» 

3hi2m 

3h20"' 

3h  28'^ 

Shs6^ 

3^44^ 

3^52^ 

These 

IN   — >■ 

^r-       IN 

After- 

8 56 

8   48 

8   40 

8  32 

8  24 

8   16 

8     8 

After- 

noon 

noon 

0° 

7193 

7432 

7661 

7879 

8091 

8290 

8480 

0° 

2 

7190 

7427 

7656 

7875 

8085 

8285 

8476 

2 

4 

7176 

7413 

7642 

7861 

8071 

8269 

8461 

4 

6 

7153 

7391 

7619 

7836 

8046 

8245 

8433 

6 

8 

7124 

7358 

7586 

7803 

8011 

8210 

8399 

8 

10 

7084 

7318 

7544 

7761 

7968 

8164 

8352 

10 

12 

7036 

7269 

7494 

7707 

7914 

8110 

8284 

12 

14 

6979 

7211 

7433 

7645 

7850 

8044 

8228 

14 

16 

6915 

7144 

7364 

7575 

7776 

7969 

8153 

16 

18 

6841 

7068 

7286 

7494 

7695 

7884 

8065 

18 

20 

6759 

6984 

7197 

7404 

7603 

7791 

7969 

20 

22 

6670 

6890 

7103 

7307 

7501 

7686 

7863 

22 

24 

6572 

6789 

6998 

7199 

7391 

7573 

7749 

24 

26 

6466 

6679 

6885 

7082 

7271 

7450 

7623 

26 

28 

6352 

6561 

6764 

6958 

7144 

7319 

7489 

28 

30 

6230 

6436 

6634 

6825 

7006 

7180 

7345 

30 

32 

6101 

6302 

6497 

6683 

6861 

7031 

7191 

32 

34 

5964 

6160 

6351 

6533 

6707 

6873 

7031 

34 

36 

5820 

6012 

6197 

6375 

6545 

6707 

6861 

36 

m 

38 

5669 

5856 

6037 

6210 

6375 

6533 

6683 

38 

o 

40 

5511 

5693 

5868 

6037 

6197 

6351 

6497 

40 

00 

H 

42 

5346 

5522 

5693 

5856 

6012 

6160 

6302 

42 

Q 

< 

44 

5175 

5346 

5511 

5669 

5820 

5964 

6101 

44 

h3 

M 
J 

46 

4997 

5162 

5321 

5474 

5619 

5758 

5891 

46 

H 

48 

4812 

4973 

5126 

5272 

5414 

5547 

5674 

48 

< 

Q 

50 

4624 

4776 

4923 

5065 

5200 

5328 

5451 

50 

52 

4429 

4575 

4717 

4852 

4980 

5104 

5222 

52 

54 

4229 

4368 

4503 

4633 

4755 

4873 

4984 

54 

56 

4023 

4156 

4284 

4407 

4524 

4635 

4742 

56 

58 

3812 

3939 

4060 

4176 

4287 

4393 

4493 

58 

60 

3596 

3715 

3830 

3941 

4044 

4145 

4240 

60 

62 

3378 

3489 

3596 

3700 

3798 

3892 

3981 

62 

64 

3154 

3258 

3358 

3454 

3546 

3634 

3718 

64 

66 

2925 

3023 

3116 

3205 

3291 

3372 

3449 

66 

68 

2695 

2784 

2869 

2952 

3031 

3106 

3177 

68 

70 

2460 

2542 

2620 

2696 

2767 

2835 

2900 

70 

72 

2223 

2297 

2367 

2435 

2500 

2563 

2620 

72 

74 

1983 

2048 

2112 

2172 

2230 

2285 

2337 

74 

76 

1740 

1798 

1853 

1906 

1957 

2005 

2051 

76 

78 

1496 

1545 

1592 

1639 

1682 

1724 

1763 

78 

80 

1249 

1290 

1330 

1369 

1405 

1440 

1472 

80 

82 

1001 

1034 

1066 

1096 

1126 

1153 

1180 

82 

84 

752 

777 

801 

824 

846 

867 

886 

84 

86 

502 

518 

534 

550 

564 

578 

592 

86 

88 

251 

259 

267 

275 

282 

289 

296 

88 

Use 

Use 

These 

46° 

48° 

50° 

52° 

54° 

56° 

58° 

These 

IN  —> 

"< —    IN 

Fore- 

134 

132 

130 

128 

126 

124 

122 

Fore- 

noon 

noon 

Use 

Use 

These 

226° 

228° 

230° 

232° 

234° 

236° 

238° 

These 

IN   — >• 

■< —    IN 

After- 

314 

312 

310 

308 

306 

304 

302 

After- 

noon 

1 
1 

'rue  Be^ 

i.RING  OR 

Azimuth 

[ 

noon 

288 


Table  11.    Azimuth 


T, 

The  Shi 

p's  Apparent  Time  for  a  Sun  Observation, 

or  TH] 

a  HouR-j 

\.ngle  fc 

R  A  StAJ 

B  Obseri 

/^ATION 

Use 

Use 

These 

l^h    o'" 

16h    8^ 

16^16^ 

16^  24"^ 

16^32^ 

16>^  40"" 

16^48^ 

lef^se"" 

These 

IN 

^ 

■<— 

IN 

Fore- 

^0  0 

19   52 

19   U 

19   36 

19   28 

19   20 

19   12 

19     4 

Fore- 

noon 

noon 

Use 

Use 

These 

^A  Qm 

4^    8^ 

^hiem 

4^24"" 

^hs^m 

4^40"^ 

4^  48"^ 

4^56^ 

These 

IN 

—> 

^ 

IN 

After- 

8    0 

7  62 

7   U 

7  36 

7  28 

7  20 

7    12 

7     4 

After- 

noon 

noon 

0° 

8660 

8828 

8989 

9135 

9272 

9397 

9510 

9612 

0° 

2 

8656 

8824 

8982 

9131 

9266 

9391 

9506 

9607 

2 

4 

8640 

8808 

8966 

9114 

9249 

9374 

9486 

9590 

4 

6 

8612 

8780 

8939 

9084 

9221 

9346 

9458 

9561 

6 

8 

8576 

8744 

8900 

9046 

9181 

9305 

9419 

9519 

8 

10 

8529 

8696 

8851 

8997 

9131 

9253 

9367 

9466 

10 

12 

8470 

8636 

8792 

8935 

9069 

9191 

9303 

9401 

12 

14 

8403 

8567 

8722 

8863 

8997 

9118 

9228 

9326 

14 

16 

8326 

8487 

8640 

8782 

8913 

9033 

9143 

9241 

16 

18 

8235 

8397 

8549 

8688 

8818 

8937 

9044 

9143 

i8 

20 

8137 

8296 

8447 

8584 

8714 

8831 

8937 

9033 

20 

22 

8030 

8187 

8333 

8470 

8596 

8714 

8818 

8913 

22 

24 

7912 

8067 

8212 

8347 

8470 

8584 

8688 

8782 

24 

26 

7784 

7936 

8078 

8212 

8333 

8447 

8549 

8640 

26 

28 

7647 

7796 

7936 

8067 

8187 

8296 

8397 

8487 

28 

30 

7501 

7647 

7784 

7912 

8030 

8137 

8235 

8326 

30 

32 

7345 

7489 

7623 

7749 

7863 

7969 

8065 

8153 

32 

34 

7180 

7319 

7450 

7573 

7686 

7791 

7884 

7969 

34 

36 

7006 

7144 

7271 

7391 

7501 

7603 

7695 

7776 

36 

95 

38 

6825 

6958 

7082 

7199 

7307 

7404 

7494 

7575 

38 

o 

40 

6634 

6764 

6885 

6998 

7103 

7197 

7286 

7364 

40 

to 

H 

42 

6436 

6561 

6679 

6789 

6890 

6984 

7068 

7144 

42 

Q 

•< 

Iz; 

44 

6230 

6352 

6466 

6572 

6670 

6759 

6841 

6915 

44 

0 

M 

46 

6016 

6134 

6243 

6346 

6440 

6528 

6607 

6677 

46 

H 
J 

48 

5794 

5907 

6015 

6212 

6204 

6288 

6363 

6433 

48 

< 

Q 

50 

5567 

5675 

5777 

5872 

5960 

6040 

6113 

6179 

50 

52 

5332 

5437 

5534 

5624 

5709 

5785 

5856 

5919 

52 

54 

5091 

5190 

5283 

5370 

5450 

5524 

5590 

5650 

54 

56 

4843 

4938 

5025 

5109 

5184 

5255 

5319 

5375 

56 

58 

4588 

4678 

4763 

4841 

4914 

4979 

5040 

5094 

58 

60 

4330 

4415 

4493 

4567 

4635 

4699 

4755 

4806 

60 

62 

4065 

4145 

4220 

4288 

4353 

4412 

4465 

4513 

62 

64 

3796 

3871 

3941 

4005 

4064 

4119 

4170 

4214 

64 

66 

3522 

3591 

3656 

3715 

3771 

3821 

3868 

3910 

66 

68 

3244 

3308 

3367 

3422 

3473 

3521 

3563 

3600 

68 

70 

2962 

3020 

3074 

3125 

3171 

3214 

3253 

3288 

70 

72 

2676 

2729 

2777 

2823 

2865 

2903 

2939 

2970 

72 

74 

2387 

2434 

2477 

2519 

2556 

2590 

2622 

2650 

74 

76 

2095 

2136 

2174 

2210 

2244 

2274 

2301 

2326 

76 

78 

1801 

1835 

1868 

1900 

1928 

1954 

1977 

1998 

78 

80 

1504 

1533 

1561 

1586 

1610 

1632 

1651 

1669 

80 

82 

1205 

1228 

1251 

1272 

1290 

1308 

1323 

1338 

82 

84 

905 

923 

940 

955 

969 

982 

994 

1005 

84 

86 

604 

616 

627 

637 

647 

656 

663 

671 

86 

88 

302 

308 

314 

319 

324 

328 

332 

336 

88 

Use 

Use 

These 

60° 

62° 

64° 

66° 

68° 

70° 

72° 

74° 

These 

IN 

> 

<— 

IN 

Fore- 

120 

118 

116 

114 

112 

110 

108 

106 

Fore- 

noon 

noon 

Use 

Use 

These 

240° 

242° 

244° 

246° 

248° 

250° 

252° 

254° 

These 

IN 

— > 

^ 

IN 

After- 

300 

298 

296 

294 

292 

290 

288 

286 

After- 

noon 

True 

Bearin 

Q  OR  Azi 

MUTH 

noon 

Table  11.    Azimuth 


289 


T, 

The  Ship's  Apparent  T 

[ME  FOR 

A  Sun  Observation, 

OR  th 

e  Hour- 

angle  f 

OR  A  StA 

R  Obser 

vation 

Use 

Use 

These 

17h   4"- 

17^12^ 

17^20^ 

17^28^ 

17h36»t 

17^44^ 

17h52^ 

tgh       Qm 

These 

IN 

—>■ 

<— 

IN 

Fore- 

18 56 

18   48 

18   40 

18   32 

18   24 

18    16 

18     8 

18      0 

Fore- 

noon 

► 

noon 

Use 

Use 

These 

6^   4"* 

Sh  12"" 

5^20^ 

5f'28^ 

5^36"" 

5^44"^ 

5^52^ 

Qh       Qm 

These 

IN 

—> 

-<— 

IN 

After- 

6 56 

6   48 

6   40 

6  82 

6  24 

6    16 

6     8 

6      0 

After- 

noon 

noon 

0° 

9703 

9781 

9849 

9904 

9945 

9974 

9993 

10000 

0° 

2 

9696 

9774 

9842 

9897 

9940 

9970 

9988 

9993 

2 

4 

9679 

9757 

9824 

9879 

9922 

9951 

9970 

9974 

4 

6 

9649 

9727 

9795 

9849 

9891 

9922 

9940 

9945 

6 

8 

9610 

9687 

9752 

9806 

9849 

9879 

9897 

9904 

8 

10 

9557 

9634 

9699 

9752 

9795 

9824 

9842 

9849 

10 

12 

9491 

9568 

9634 

9687 

9727 

9757 

9774 

9781 

12 

14 

9414 

9491 

9557 

9610 

9649 

9679 

9696 

9703 

14 

16 

9326 

9401 

9466 

9519 

9561 

9590 

9607 

9612 

16 

18 

9228 

9303 

9367 

9419 

9458 

9486 

9506 

9510 

18 

20 

9118 

9191 

9253 

9305 

9346 

9374 

9391 

9397 

20 

22 

8997 

9069 

9131 

9181 

9221 

9249 

9266 

9272 

22 

24 

8863 

8935 

8997 

9046 

9084 

9114 

9131 

9135 

24 

26 

8722 

8792 

8851 

8900 

8939 

8966 

8982 

8989 

26 

28 

8567 

8636 

8696 

8744 

8780 

8808 

8824 

8828 

28 

30 

8403 

8470 

8529 

8576 

8612 

8640 

8656 

8660 

30 

32 

8228 

8284 

8352 

8399 

8433 

8461 

8476 

8480 

32 

34 

8044 

8110 

8164 

8210 

8245 

8269 

8285 

8290 

34 

36 

7850 

7914 

7968 

8011 

8046 

8071 

8085 

8091 

36 

05 

38 

7645 

7707 

7761 

7803 

7836 

7861 

7875 

7879 

38 

o 

40 

7433 

7494 

7544 

7586 

7619 

7642 

7656 

7661 

40 

m 

■4> 

42 

7211 

7269 

7318 

7358 

7391 

7413 

7427 

7432 

42 

Q 

z 

44 

6979 

7036 

7084 

7124 

7153 

7176 

7190 

7193 

44 

3 

46 

6741 

6795 

6841 

6879 

6908 

6929 

6942 

6947 

46 

g 

a 
Q 

48 

6492 

6545 

6591 

6627 

6655 

6676 

6688 

6691 

48 

< 

50 

6237 

6288 

6330 

6365 

6393 

6412 

6424 

6428 

50 

52 

5973 

6022 

6063 

6096 

6124 

6142 

6153 

6156 

52 

54 

5703 

5749 

5789 

5821 

5845 

5864 

5875 

5878 

54 

56 

5426 

5470 

5507 

5538 

5562 

5578 

5589 

5593 

56 

58 

5141 

5183 

5219 

5248 

5270 

5286 

5297 

5299 

58 

60 

4852 

48^1 

4923 

4951 

4973 

4987 

4997 

5000 

60 

62 

4555 

4592 

4624 

4649 

4669 

4684 

4692 

4695 

62 

64 

4253 

4287 

4317 

4341 

4360 

4373 

4381 

4383 

64 

66 

3947 

3978 

4005 

4028 

4044 

4058 

4065 

4067 

66 

68 

3635 

3664 

3689 

3710 

3726 

3737 

3744 

3746 

68 

70 

3319 

3346 

3368 

3387 

3402 

3412 

3418 

3421 

70 

72 

2998 

3023 

3043 

3060 

3073 

3082 

3088 

3090 

72 

74 

2674 

2696 

2714 

2730 

2742 

2750 

2755 

2756 

74 

76 

2347 

2367 

2382 

2395 

2406 

2413 

2417 

2419 

76 

78 

2018 

2033 

2047 

2059 

2067 

2074 

2078 

2079 

78 

80 

1685 

1698 

1710 

1720 

1726 

1732 

1736 

1737 

80 

82 

1351 

1361 

1371 

1378 

1384 

1389 

1391 

1392 

82 

84 

1015 

1022 

1029 

1035 

1040 

1042 

1045 

1045 

84 

86 

677 

682 

687 

691 

694 

696 

697 

698 

86 

88 

339 

341 

344 

346 

347 

348 

349 

349 

88 

Use 

Use 

These 

76° 

78° 

80° 

82° 

84° 

86° 

88° 

90° 

These 

IN  • 

— >• 

^ 

in 

Fore- 

104 

102 

100 

98 

96 

94 

92 

90 

Fore- 

noon 

noon 

Use 

Use 

These 

256° 

258° 

260° 

262° 

264° 

266° 

268° 

270° 

These 

IN 

^ 

<— 

IN 

After- 
noon 

284 

282 

280 
True 

278 
Bearin 

276 

G  OR  Azi 

274 

muth 

272 

270 

After- 
noon 

290 


Table  12.    Auxiliary  Azimuth  Table 


Declinations 

Latitude 

0° 

2° 

4° 

6° 

8° 

10° 

12° 

14° 

16° 

18° 

20° 

22° 

24° 

0° 

0° 

2 

0 

90° 

4 

0 

30 

90° 

6 

0 

20 

42 

90° 

8 

0 

15 

30 

49 

90° 

10 

0 

12 

24 

37 

53 

90° 

12 

0 

10 

20 

30 

42 

57 

90° 

14 

0 

8 

17 

26 

35 

46 

59 

90° 

16 

0 

7 

15 

22 

30 

39 

49 

01 

90° 

18 

0 

6 

13 

20 

27 

34 

42 

52 

63 

90° 

20 

0 

6 

12 

18 

24 

31 

37 

45 

54 

65 

90° 

22 

0 

5 

11 

16 

22 

28 

34 

40 

47 

56 

66 

90° 

24 

0 

5 

10 

15 

20 

25 

31 

36 

43 

49 

57 

67 

90° 

26 

0 

5 

9 

14 

19 

23 

28 

34 

39 

45 

51 

59 

68 

28 

0 

4 

9 

13 

17 

22 

26 

31 

36 

41 

47 

53 

60 

30 

0 

4 

8 

12 

16 

20 

25 

29 

33 

38 

43 

49 

54 

32 

0 

4 

8 

11 

15 

19 

23 

27 

31 

36 

40 

45 

50 

34 

0 

4 

7 

11 

14 

18 

22 

26 

30 

34 

38 

42 

47 

36 

0 

3 

7 

10 

14 

17 

21 

24 

28 

32 

36 

40 

44 

38 

0 

3 

7 

10 

13 

16 

20 

23 

27 

30 

34 

37 

41 

40 

0 

3 

6 

9 

12 

16 

19 

22 

25 

29 

32 

36 

39 

42 

0 

3 

6 

9 

12 

15 

18 

21 

24 

28 

31 

34 

37 

44 

0 

3 

6 

9 

12 

14 

17 

20 

23 

26 

30 

33 

36 

46 

0 

3 

6 

8 

11 

14 

17 

20 

23 

25 

28 

31 

34 

48 

0 

3 

5 

8 

11 

14 

16 

19 

22 

25 

27 

30 

33 

50 

0 

3 

5 

8 

10 

13 

16 

18 

21 

24 

27 

29 

32 

52 

0 

3 

5 

8 

10 

13 

15 

18 

20 

23 

26 

28 

31 

54 

0 

2 

5 

7 

10 

12 

15 

17 

20 

22 

25 

28 

30 

56 

0 

2 

5 

7 

10 

12 

15 

17 

19 

22 

24 

27 

29 

58 

0 

2 

5 

7 

9 

12 

14 

17 

19 

21 

24 

26 

29 

60 

0 

2 

5 

7 

9 

12 

14 

16 

19 

21 

23 

26 

28 

Table  12.    Completed 


Declinations 

Latitude 

26° 

28° 

30° 

32° 

34° 

36° 

38° 

40° 

42° 

44° 

46° 

48° 

50° 

26° 

90° 

28 

69 

90° 

30 

61 

70 

90° 

32 

56 

62 

71 

90° 

34 

52 

57 

63 

71 

90° 

36 

48 

53 

58 

64 

72 

90° 

38 

45 

50 

54 

59 

65 

73 

90° 

40 

43 

47 

51 

56 

60 

66 

73 

90° 

42 

41 

45 

48 

53 

57 

61 

67 

74 

90° 

44 

39 

43 

46 

50 

54 

58 

62 

68 

74 

90° 

46 

38 

41 

44 

47 

51 

55 

59 

63 

68 

75 

90° 

48 

36 

39 

42 

45 

49 

52 

56 

60 

64 

69 

75 

90" 

50 

35 

38 

41 

44 

47 

50 

53 

57 

61 

65 

70 

76 

90° 

52 

34 

37 

39 

42 

45 

48 

51 

55 

58 

62 

66 

71 

76 

54 

33 

35 

38 

41 

44 

47 

50 

53 

56 

59 

63 

67 

71 

56 

32 

34 

37 

40 

42 

45 

48 

51 

54 

57 

60 

64 

68 

58 

31 

33 

36 

39 

41 

44 

47 

49 

52 

55 

58 

61 

65 

60 

30 

33 

35 

38 

40 

43 

45 

48 

51 

53 

56 

59 

62 

292 


Table  13.    Kelvin's  Sumner  Line  Table 


b 

3 

I  = 

0° 

a 

= 

1° 

a  = 

2° 

a  = 

3° 

a  = 

4° 

a  = 

5° 

a  =  6° 

K 

Q 

K 

Q 

K 

Q 

o  / 

K 

Q 

K 

Q 

K 

Q 

K 

Q 

o 

o 

/ 

o  / 

o 

/ 

O   ' 

o   / 

o   / 

o  / 

o  / 

o  / 

o  / 

o  / 

o   / 

o  / 

0 

0 

0 

0  0 

0 

0 

1  0 

0  0 

2  0 

0  0 

3  0 

0  0 

4  0 

0  0 

5  0 

0  0 

6  0 

1 

1 

0 

0 

1 

0 

0 

1  0 

0 

1  0 

0 

1  0 

0 

1  0 

0 

1  0 

0 

2 

2 

0 

0 

2 

0 

0 

2  0 

0 

2  0 

0 

2  0 

0 

2  0 

0 

59 

0 

3 

3 

0 

0 

3 

0 

0 

3  0 

0 

3  0 

0 

3  0 

0 

59 

0 

2  59 

0 

4 

4 

0 

0 

4 

0 

0 

4  0 

0 

4  0 

0 

59 

1 

3  59 

1 

3  59 

1 

5 

5 

0 

0 

5 

0 

0 

5  0 

0 

5  0 

1 

4  59 

1 

4  59 

1 

4  58 

1 

6 

6 

0 

0 

6 

0 

0 

6  0 

1 

6  0 

1 

5  59 

1 

5  59 

2 

5  58 

2 

7 

7 

0 

0 

7 

0 

0 

7  0 

1 

59 

1 

6  59 

2 

6  58 

2 

6  58 

3 

8 

8 

0 

0 

8 

0 

1 

8  0 

1 

7  59 

2 

7  59 

2 

7  58 

3 

7  57 

4 

9 

9 

0 

0 

9 

0 

1 

9  0 

1 

8  59 

2 

8  59 

3 

8  58 

4 

8  57 

5 

10 

10 

0 

0 

10 

0 

1 

10  0 

2 

9  59 

3 

9  59 

4 

9  58 

5 

9  57 

6 

11 

11 

0 

0 

11 

0 

1 

11  0 

2 

10  59 

3 

10  58 

4 

10  57 

6 

10  56 

7 

12 

12 

0 

0 

12 

0 

1 

12  0 

3 

11  59 

4 

11  58 

5 

11  57 

7 

11  56 

8 

13 

13 

0 

0 

13 

0 

2 

13  0 

3 

12  59 

5 

12  58 

6 

12  57 

8 

12  56 

9 

14 

14 

0 

0 

14 

0 

2 

59 

4 

13  59 

5 

13  58 

7 

13  57 

9 

13  55 

11 

15 

15 

0 

0 

15 

0 

2 

14  59 

4 

14  59 

6 

14  58 

8 

14  56 

10 

14  55 

13 

16 

16 

0 

0 

16 

0 

2 

15  59 

5 

15  59 

7 

15  58 

10 

15  56 

12 

15  55 

14 

17 

17 

0 

0 

17 

0 

3 

16  59 

5 

16  59 

8 

16  57 

11 

16  56 

14 

16  54 

16 

18 

18 

0 

0 

18 

0 

3 

17  59 

6 

17  58 

9 

17  57 

12 

17  56 

15 

17  54 

18 

19 

19 

0 

0 

19 

0 

3 

18  59 

7 

18  58 

10 

18  57 

14 

18  55 

17 

18  54 

21 

20 

20 

0 

0 

20 

0 

4 

19  59 

8 

19  58 

11 

19  57 

15 

19  55 

19 

19  53 

23 

21 

21 

0 

0 

21 

0 

4 

20  59 

9 

20  58 

13 

20  57 

17 

20  55 

21 

20  53 

25 

22 

22 

0 

0 

22 

0 

5 

21  59 

9 

21  58 

14 

21  57 

19 

21  55 

23 

21  52 

28 

23 

23 

0 

0 

23 

0 

5 

22  59 

10 

22  58 

15 

22  56 

21 

22  54 

26 

22  52 

31 

24 

24 

0 

0 

24 

0 

6 

23  59 

11 

23  58 

17 

23  56 

23 

23  54 

28 

23  52 

34 

25 

25 

0 

0 

25 

0 

6 

24  59 

12 

24  58 

18 

24  56 

25 

24  54 

31 

24  51 

37 

26 

26 

0 

0 

26 

0 

7 

25  59 

13 

25  58 

20 

25  56 

27 

25  54 

34 

25  51 

40 

27 

27 

0 

0 

27 

0 

7 

26  59 

15 

26  58 

22 

26  56 

29 

26  53 

37 

26  50 

44 

28 

28 

0 

0 

28 

0 

8 

27  59 

16 

27  57 

24 

27  56 

32 

27  53 

40 

27  50 

47 

29 

29 

0 

0 

29 

0 

9 

28  59 

17 

28  57 

26 

28  55 

34 

28  53 

43 

28  50 

51 

30 

30 

0 

0 

30 

0 

9 

29  59 

19 

29  57 

28 

29  55 

37 

29  52 

46 

29  49 

55 

31 

31 

0 

0 

31 

0 

10 

30  59 

20 

30  57 

30 

30  55 

40 

30  52 

50 

30  49 

59 

32 

32 

0 

0 

32 

0 

11 

31  59 

21 

31  57 

32 

31  55 

43 

31  52 

53 

3148 

7  4 

33 

33 

0 

0 

33 

0 

12 

32  59 

23 

32  57 

34 

32  55 

46 

32  52 

57 

32  48 

9 

34 

34 

0 

0 

34 

0 

12 

33  59 

25 

33  57 

37 

33  54 

49 

33  51 

6  1 

33  47 

14 

35 

35 

0 

0 

35 

0 

13 

34  59 

26 

34  57 

40 

34  54 

53 

34  51 

6 

34  47 

19 

36 

36 

0 

0 

36 

0 

14 

35  58 

■28 

35  57 

42 

35  54 

56 

35  51 

10 

35  46 

24 

37 

37 

0 

0 

37 

0 

15 

36  58 

30 

36  56 

45 

36  54 

5  0 

36  50 

15 

36  46 

30 

38 

38 

0 

0 

38 

0 

16 

37  58 

32 

37  56 

48 

37  53 

4 

37  50 

20 

37  45 

36 

39 

39 

0 

0 

39 

0 

17 

38  58 

34 

38  56 

51 

38  53 

8 

38  49 

25 

38  45 

42 

40 

40 

0 

0 

40 

0 

18 

39  58 

37 

39  56 

55 

39  53 

13 

39  49 

31 

39  44 

49 

41 

41 

0 

0 

41 

0 

19 

40  58 

39 

40  56 

58 

40  53 

18 

40  49 

37 

40  44 

56 

42 

42 

0 

0 

42 

0 

21 

41  58 

41 

41  56 

4  2 

41  52 

23 

4148 

43 

41  43 

8  3 

43 

43 

0 

0 

43 

0 

22 

42  58 

44 

42  56 

6 

42  52 

28 

42  48 

49 

42  42 

11 

44 

44 

0 

0 

59 

23 

43  58 

47 

43  55 

10 

43  52 

33 

43  47 

56 

43  42 

19 

45 

45 

0 

0 

44  59 

25 

44  58 

50 

44  55 

14 

44  52 

39 

44  47 

7  3 

44  41 

27 

Table  13.    Kelvin's  Sumner  Line  Table 


293 


b 

a 

= 

0° 

a  = 

1° 

a  =  2° 

a  = 

=  3° 

a  = 

:  4° 

a  = 

=  5° 

a  =  6° 

K 

Q 

0  / 

K 

Q 

K 

Q 

K 

o   / 

Q 

K 

Q 

K 

Q 

K 

Q 

o 

o 

) 

o   / 

o   / 

o   / 

o    / 

O    ' 

o   / 

o   / 

o   / 

o   / 

o   / 

o  / 

45 

45 

0 

00 

44  59 

1  25 

44  58 

2  50 

44  55 

4  14 

14  52 

5  39 

44  47 

7  3 

44  41 

8  27 

46 

46 

0 

0 

45  59 

26 

45  58 

53 

45  55 

19 

45  51 

45 

45  46 

11 

45  41 

36 

47 

47 

0 

0 

46  59 

28 

46  58 

56 

46  55 

24 

46  51 

51 

46  46 

19 

46  40 

46 

48 

48 

0 

0 

47  59 

30 

47  58 

59 

47  55 

29 

47  51 

58 

47  45 

27 

47  39 

56 

49 

49 

0 

0 

48  59 

31 

48  58 

3  3 

48  55 

34 

48  50 

6  5 

48  45 

36 

48  38 

9  6 

50 

50 

0 

0 

49  59 

33 

49  58 

7 

49  54 

40 

49  50 

13 

49  44 

45 

49  38 

17 

51 

51 

0 

0 

50  59 

35 

50  57 

11 

50  54 

46 

50  50 

21 

50  44 

55 

50  37 

29 

52 

52 

0 

0 

51  59 

37 

51  57 

15 

51  54 

52 

51  49 

29 

51  43 

8  5 

51  36 

41 

53 

53 

0 

0 

52  59 

40 

52  57 

19 

52  54 

59 

52  49 

38 

52  43 

10 

52  35 

54 

54 

54 

0 

0 

53  59 

42 

53  57 

24 

53  54 

5  6 

53  49 

47 

53  42 

28 

53  34 

10  8 

55 

55 

0 

0 

54  59 

45 

54  57 

29 

54  53 

13 

54  48 

57 

54  41 

40 

54  33 

23 

56 

56 

0 

0 

55  59 

47 

bo  bl 

34 

55  53 

21 

55  48 

7  8 

55  41 

53 

55  32 

39 

57 

57 

0 

0 

56  59 

50 

56  57 

40 

56  53 

30 

56  47 

19 

56  40 

9  7 

56  31 

55 

58 

58 

0 

0 

57  59 

53 

57  57 

46 

57  52 

39 

57  47 

31 

57  39 

22 

57  30 

11  13 

59 

59 

0 

0 

58  59 

56 

58  57 

53 

58  52 

49 

58  46 

44 

58  38 

38 

58  29 

32 

60 

60 

0 

0 

59  59 

2  0 

59  56 

4  0 

59  52 

59 

59  46 

58 

59  37 

56 

59  28 

52 

61 

61 

0 

0 

60  59 

4 

60  56 

7 

60  52 

6  10 

60  45 

8  13 

60  36 

10  14 

60  26 

12  14 

62 

62 

0 

0 

61  59 

8 

61  56 

15 

61  51 

22 

61  44 

28 

61  35 

33 

61  25 

37 

6.3 

63 

0 

0 

62  59 

12 

62  56 

24 

62  51 

35 

62  44 

45 

62  34 

54 

62  23 

13  2 

64 

64 

0 

0 

63  59 

17 

63  56 

33 

63  50 

49 

63  43 

9  4 

63  33 

11  17 

63  22 

29 

65 

65 

0 

0 

64  59 

22 

64  56 

43 

64  50 

7  4 

64  42 

24 

64  32 

42 

64  20 

58 

66 

66 

0 

0 

65  59 

27 

65  55 

54 

65  49 

20 

65  41 

45 

65  31 

12  8 

65  18 

14  29 

67 

67 

0 

0 

66  59 

33 

66  55 

5  6 

66  49 

38 

66  40 

10  9 

66  29 

37 

66  16 

15  3 

68 

68 

0 

0 

67  59 

40 

67  55 

19 

67  48 

58 

67  39 

34 

67  28 

13  9 

67  14 

40 

69 

69 

0 

0 

68  59 

47 

68  55 

34 

68  48 

8  19 

68  38 

11  2 

68  26 

43 

68  12 

16  21 

70 

70 

0 

0 

69  59 

55 

69  54 

•50 

69  47 

43 

69  37 

33 

69  25 

14  21 

69  9 

17  5 

71 

71 

0 

0 

70  58 

3  4 

70  54 

6  7 

70  46 

9  9 

70  36 

12  7 

70  23 

15  2 

70  6 

54 

72 

72 

0 

0 

71  58 

14 

71  54 

27 

7146 

38 

71  35 

45 

71  20 

48 

71  3 

18  47 

73 

73 

0 

0 

72  58 

25 

72  53 

49 

72  45 

10  10 

72  33 

13  27 

72  18 

16  40 

72  0 

19  46 

74 

74 

0 

0 

73  58 

37 

73  53 

7  13 

73  44 

46 

73  31 

14  14 

73  15 

17  37 

56 

20  52 

75 

75 

0 

0 

74  58 

51 

74  52 

41 

74  43 

11  27 

74  29 

15  7 

74  12 

18  41 

73  52 

22  6 

76 

76 

0 

0 

75  58 

4  8 

75  52 

8  13 

75  41 

12  13 

75  27 

16  7 

75  9 

19  53 

74  47 

23  29 

77 

77 

0 

0 

76  58 

26 

76  51 

50 

76  40 

13  7 

76  25 

17  16 

76  5 

21  15 

75  42 

25  3 

78 

78 

0 

0 

77  58 

48 

77  50 

9  32 

77  38 

14  9 

77  22 

18  35 

77  1 

22  49 

76  36 

26  49 

79 

79 

0 

0 

78  57 

5  14 

78  49 

10  22 

78  36 

15  22 

78  18 

20  8 

56 

24  38 

77  29 

28  51 

80 

80 

0 

0 

79  57 

44 

79  48 

11  22 

79  34 

16  48 

79  14 

21  56 

78  50 

26  44 

78  21 

31  11 

81 

81 

0 

0 

80  57 

6  22 

80  47 

12  35 

80  31 

18  31 

80  9 

24  5 

79  43 

29  13 

79  12 

33  54 

82 

82 

0 

0 

81  56 

7  9 

81  45 

14  5 

81  28 

20  38 

81  4 

26  41 

80  34 

32  9 

80  1 

37  4 

83 

83 

0 

0 

82  56 

8  9 

82  43 

15  59 

82  23 

23  16 

57 

29  51 

81  24 

35  40 

47 

40  47 

84 

84 

0 

0 

83  55 

9  29 

83  41 

18  28 

83  18 

26  38 

82  48 

33  47 

82  12 

39  56 

81  31 

45  9 

85 

85 

0 

0 

84  54 

11  20 

84  37 

21  50 

84  10 

31  1 

83  36 

38  44 

56 

45  7 

82  12 

50  20 

86 

86 

0 

0 

85  53 

14  3 

85  32 

26  36 

85  0 

36  55 

84  21 

45  4 

83  36 

51  26 

48 

56  26 

87 

87 

0 

0 

86  50 

18  27 

86  24 

33  43 

45 

45  2 

85  0 

53  11 

84  10 

59  7 

83  18 

63  32 

88 

88 

0 

0 

87  46 

26  34 

87  10 

45  1 

86  24 

56  20 

32 

63  29 

37 

68  15 

41 

71  38 

89 

89 

0 

0 

88  35 

45  0 

46 

63  27 

50 

71  35 

53 

75  59 

54 

78  43 

55 

80  34 

90 

90 

0 

0 

89  0 

90  0 

88  0 

90  0 

87  0 

90  0 

86  0 

90  0 

85  0 

90  0 

84  0 

90  0 

294 


Table  13.    Kelvin's  Sumner  Line  Table 


b 

0 

a  = 

=  7° 

a  = 

8° 

a  = 

=  9° 

a  = 

10° 

a  = 

11° 

a  = 

13° 

a  = 

13° 

K 

Q 

K 

Q 

K 

Q 

K 

Q 

K 

Q 

K 

Q 

K 

Q 

O    ' 

0   ' 

O     ' 

o   / 

o   / 

o   / 

o   / 

o   / 

o   / 

O    ' 

o   / 

o  / 

o  / 

o   / 

0 

0  0 

7  0 

0  0 

8  0 

0  0 

9  0 

0  0 

10  0 

0  0 

11  0 

0  0 

12  0 

0  0 

13  0 

1 

1  0 

0 

59 

0 

59 

0 

59 

0 

59 

0 

59 

0 

58 

0 

2 

59 

0 

1  59 

0 

1  59 

0 

1  58 

0 

1  58 

0 

1  57 

0 

1  57 

0 

3 

2  59 

1 

2  58 

1 

2  58 

1 

2  57 

1 

2  57 

1 

2  56 

1 

2  55 

1 

4 

3  58 

1 

3  58 

1 

3  57 

1 

3  56 

1 

3  56 

2 

3  55 

2 

3  54 

2 

5 

4  58 

2 

4  57 

2 

4  56 

2 

4  55 

2 

4  54 

3 

4  53 

3 

4  52 

3 

6 

5  57 

2 

5  56 

3 

5  56 

3 

5  55 

3 

5  53 

4 

5  52 

4 

5  51 

4 

7 

6  57 

3 

6  56 

4 

6  55 

4 

6  54 

4 

6  52 

5 

6  51 

5 

6  49 

6 

8 

7  56 

4 

7  55 

5 

7  54 

5 

7  53 

6 

7  51 

6 

7  49 

7 

7  48 

8 

9 

8  56 

5 

8  55 

6 

8  53 

7 

8  52 

7 

8  50 

8 

8  48 

9 

8  46 

10 

10 

9  55 

6 

9  54 

7 

9  53 

8 

9  51 

9 

9  49 

10 

9  47 

11 

944 

12 

11 

10  55 

8 

10  53 

9 

10  52 

10 

10  50 

11 

10  48 

12 

10  45 

13 

10  43 

14 

12 

11  55 

9 

11  53 

11 

11  51 

12 

11  49 

13 

11  47 

14 

11  44 

16 

11  41 

17 

13 

12  54 

11 

12  52 

13 

12  50 

14 

12  48 

15 

12  45 

17 

12  43 

19 

12  40 

20 

14 

13  54 

13 

13  52 

15 

13  49 

16 

13  47 

18 

13  44 

20 

13  41 

22 

13  38 

23 

15 

14  53 

15 

14  51 

17 

14  49 

19 

14  46 

21 

14  43 

23 

14  40 

25 

14  36 

26 

16 

15  53 

17 

15  50 

19 

15  48 

21 

15  45 

24 

15  42 

26 

15  38 

28 

15  35 

30 

17 

16  52 

19 

16  50 

22 

16  47 

24 

16  44 

27 

16  41 

29 

16  37 

32 

16  33 

34 

18 

17  52 

21 

17  49 

24 

17  46 

27 

17  43 

30 

17  39 

33 

17  36 

36 

17  31 

39 

19 

18  51 

24 

18  48 

27 

18  45 

30 

18  42 

34 

18  38 

37 

18  34 

40 

18  30 

43 

20 

19  51 

27 

19  48 

30 

19  45 

34 

19  41 

38 

19  37 

41 

19  33 

45 

19  28 

48 

21 

20  50 

30 

20  47 

34 

20  44 

38 

20  40 

42 

20  36 

46 

20  31 

50 

20  26 

53 

22 

21  50 

33 

21  46 

37 

21  43 

42 

21  39 

46 

21  35 

51 

21  30 

55 

21  24 

59 

23 

22  49 

36 

22  46 

41 

22  42 

46 

22  38 

51 

22  33 

56 

22  28 

13  0 

22  23 

14  5 

24 

23  49 

39 

23  45 

45 

23  41 

50 

23  37 

56 

23  32 

12  1 

23  27 

6 

23  21 

11 

25 

24  48 

43 

24  44 

49 

24  40 

55 

24  36 

11  1 

24  31 

6 

24  25 

12 

24  19 

17 

26 

25  48 

47 

25  44 

53 

25  39 

10  0 

25  35 

6 

25  29 

12 

25  23 

18 

25  17 

24 

27 

26  47 

51 

26  43 

58 

26  38 

5 

26  33 

12 

26  28 

18 

26  22 

25 

26  15 

31 

28 

27  46 

55 

27  42 

9  3 

27  38 

10 

27  32 

18 

27  27 

25 

27  20 

32 

27  13 

39 

29 

28  46 

59 

28  41 

8 

28  37 

16 

28  31 

24 

28  25 

32 

28  18 

40 

28  11 

47 

30 

29  45 

8  4 

29  41 

13 

29  36 

22 

29  30 

31 

29  24 

39 

29  17 

48 

29  9 

56 

31 

30  45 

9 

30  40 

19 

30  35 

28 

30  29 

38 

30  22 

47 

30  15 

56 

30  7 

15  5 

32 

31  44 

14 

31  39 

25 

31  34 

35 

31  27 

45 

31  21 

55 

31  13 

14  4 

31  5 

14 

33 

32  43 

20 

32  38 

31 

32  33 

42 

32  26 

52 

32  19 

13  3 

32  11 

13 

32  3 

24 

34 

33  43 

26 

33  37 

37 

33  32 

49 

33  25 

12  0 

33  18 

12 

33  10 

23 

33  1 

34 

35 

34  42 

32 

34  37 

44 

34  30 

.  57 

34  24 

9 

34  16 

21 

34  8 

33 

59 

44 

36 

35  41 

38 

35  36 

51 

35  29 

11  5 

35  22 

18 

35  14 

31 

35  6 

43 

34  56 

55 

37 

36  41 

45 

36  35 

59 

36  28 

13 

36  21 

27 

36  13 

41 

36  4 

54 

35  54 

16  7 

38 

37  40 

52 

37  34 

10  7 

37  27 

22 

37  19 

37 

37  11 

51 

37  2 

15  6 

36  52 

20 

39 

38  39 

59 

38  33 

15 

38  26 

31 

38  18 

47 

38  9 

14  2 

38  0 

18 

37  49 

33 

40 

39  39 

9  6 

39  32 

24 

39  25 

41 

39  16 

58 

39  7 

14 

57 

31 

38  47 

46 

41 

40  38 

14 

40  31 

33 

40  23 

51 

40  15 

13  9 

40  5 

27 

39  55 

44 

39  44 

17  0 

42 

41  37 

23 

41  30 

43 

41  22 

12  2 

41  13 

21 

41  3 

40 

40  53 

58 

40  41 

15 

43 

42  36 

32 

42  29 

53 

42  21 

13 

42  12 

33 

42  1 

53 

41  51 

16  12 

41  39 

31 

44 

43  35 

41 

43  28 

11  3 

43  19 

25 

43  10 

46 

59 

15  7 

42  48 

28 

42  36 

48 

45 

44  34 

51 

44  27 

14 

44  18 

38 

44  8 

14  0 

43  57 

22 

43  46 

44 

43  33 

18  5 

Table  18.    Kelvin's  Sumner  Line  Table 


295 


b 

0 

a  =  7° 

a=8° 

a  =  9° 

a  =  10  ° 

a  =  11  ° 

a  =  13° 

a  =  13° 

K 

Q 

K 

Q 

K 

Q 

K 

Q 

K 

Q 

K 

Q 

o   / 

K 

Q 

o  / 

o  / 

o  / 

o  / 

o  / 

o   / 

o  / 

o   / 

o  / 

o   / 

o  / 

o   / 

O   ' 

45 

44  34 

9  51 

44  27 

11  14 

44  18 

12  38 

44  8 

14  0 

43  57 

15  22 

43  46 

16  44 

43  33 

18  5 

46 

45  34 

10  1 

45  26 

26 

45  16 

51 

45  6 

15 

44  55 

38 

44  43 

17  1 

44  30 

23 

47 

46  33 

12 

46  24 

39 

46  15 

13  5 

46  4 

30 

45  53 

55 

45  40 

19 

45  27 

42 

48 

47  32 

24 

47  23 

52 

47  13 

19 

47  2 

46 

46  51 

16  12 

46  38 

37 

46  24 

19  2 

49 

48  31 

36 

48  22 

12  6 

48  12 

34 

48  0 

15  3 

47  48 

30 

47  35 

57 

47  20 

23 

50 

49  30 

49 

49  20 

20 

49  10 

51 

58 

20 

48  46 

50 

48  32 

18  18 

48  17 

45 

51 

50  29 

11  2 

50  19 

35 

50  8 

14  8 

49  56 

39 

49  43 

17  10 

49  29 

40 

49  13 

20  9 

52 

51  27 

17 

51  18 

52 

51  6 

26 

50  54 

59 

50  40 

31 

50  26 

19  3 

50  9 

33 

53 

52  26 

32 

52  16 

13  9 

52  4 

45 

51  52 

16  20 

51  37 

54 

51  22 

27 

51  5 

59 

54 

53  25 

48 

53  14 

27 

53  2 

15  5 

52  49 

42 

52  34 

18  18 

52  19 

53 

52  1 

21  27 

55 

54  24 

12  5 

54  13 

46 

54  0 

26 

53  47 

17  5 

53  31 

43 

53  15 

20  20 

57 

56 

56 

55  22 

23 

55  11 

14  6 

58 

49 

54  44 

30 

54  28 

19  10 

54  11 

49 

53  53 

22  26 

57 

56  21 

42 

56  9 

28 

55  56 

16  13 

55  41 

56 

55  25 

39 

55  7 

21  19 

54  48 

58 

58 

57  19 

13  3 

57  7 

51 

56  53 

39 

56  38 

18  24 

56  21 

20  9 

56  3 

51 

55  43 

23  33 

59 

58  18 

25 

58  5 

15  16 

57  51 

17  6 

57  35 

54 

57  17 

41 

59 

22  26 

56  38 

24  9 

60 

59  16 

48 

59  3 

42 

58  48 

35 

58  32 

19  26 

58  13 

21  15 

57  54 

23  2 

57  33 

47 

61 

60  14 

14  13 

60  1 

16  10 

59  45 

18  6 

59  28 

59 

59  9 

51 

58  49 

41 

58  27 

25  28 

62 

61  12 

39 

58 

40 

60  42 

39 

60  24 

20  35 

60  5 

22  30 

59  44 

24  22 

59  21 

26  11 

63 

62  10 

15  8 

61  56 

17  12 

6139 

19  14 

61  20 

21  14 

61  0 

23  11 

60  38 

25  5 

60  15 

57 

64 

63  8 

39 

62  53 

47 

62  36 

52 

62  16 

55 

55 

55 

61  32 

52 

61  8 

27  46 

65 

64  6 

16  12 

63  50 

18  24 

63  32 

20  33 

63  12 

22  39 

62  50 

24  42 

62  26 

26  42 

62  1 

28  39 

66 

65  4 

48 

64  47 

19  4 

64  28 

21  17 

64  7 

23  26 

63  44 

25  33 

63  20 

27  36 

53 

29  35 

67 

66  1 

17  27 

65  43 

47 

65  24 

22  4 

65  2 

24  17 

64  38 

26  27 

64  13 

28  33 

63  45 

30  35 

68 

58 

18  9 

66  39 

20  34 

66  19 

55 

56 

25  12 

65  32 

27  26 

65  5 

29  34 

64  37 

31  39 

69 

67  55 

55 

67  35 

21  25 

67  14 

23  51 

66  50 

26  12 

66  25 

28  29 

57 

30  40 

65  28 

32  47 

70 

68  52 

19  45 

68  31 

22  20 

68  9 

24  51 

67  44 

27  16 

67  17 

29  37 

66  48 

31  52 

66  18 

34  1 

71 

69  48 

20  40 

69  26 

23  21 

69  3 

25  56 

68  37 

28  26 

68  9 

30  50 

67  39 

33  8 

67  7 

35  20 

72 

70  44 

21  40 

70  21 

24  27 

57 

27  8 

69  29 

29  43 

69  0 

32  10 

68  29 

34  31 

55 

36  46 

73 

7139 

22  47 

71  16 

25  40 

70  50 

28  27 

70  21 

31  6 

50 

33  37 

69  18 

36  1 

68  43 

38  18 

74 

72  34 

24  1 

72  10 

27  1 

7142 

29  53 

71  12 

32  36 

70  40 

35  12 

70  6 

37  38 

69  30 

39  57 

75 

73  29 

25  23 

73  3 

28  30 

72  34 

31  28 

72  2 

34  16 

71  28 

36  55 

53 

39  24 

70  15 

41  44 

76 

74  23 

26  55 

55 

30  9 

73  24 

33  13 

51 

36  5 

72  16 

38  47 

71  38 

41  18 

59 

43  40 

77 

75  16 

28  38 

74  46 

32  0 

74  14 

35  9 

73  39 

38  5 

73  2 

40  50 

72  23 

43  23 

71  42 

45  45 

78 

76  8 

30  34 

75  37 

34  4 

75  2 

37  18 

74  26 

40  18 

47 

43  4 

73  6 

45  38 

72  23 

48  0 

79 

59 

32  46 

76  26 

36  23 

49 

39  42 

75  11 

42  44 

74  30 

45  32 

47 

48  5 

73  2 

50  26 

80 

77  49 

35  16 

77  13 

38  59 

76  35 

42  22 

54 

45  26 

75  11 

48  13 

74  26 

50  45 

39 

53  3 

81 

78  37 

38  8 

59 

41  56 

77  18 

45  21 

76  35 

48  25 

49 

51  10 

75  2 

53  39 

74  14 

55  53 

82 

79  23 

41  25 

78  42 

45  17 

59 

48  42 

77  13 

51  43 

76  26 

54  24 

37 

56  47 

46 

58  55 

83 

80  7 

45  13 

79  23 

49  4 

78  37 

52  25 

49 

55  21 

59 

57  55 

76  8 

60  10 

75  16 

62  10 

84 

47 

49  36 

80  1 

53  22 

79  12 

56  35 

78  21 

59  20 

77  29 

6144 

36 

63  49 

42 

65  38 

85 

81  24 

54  38 

35 

58  12 

43 

61  11 

50 

63  42 

56 

65  51 

77  1 

67  42 

76  5 

69  19 

86 

57 

60  24 

81  4 

63  36 

80  9 

66  14 

79  14 

68  25 

78  18 

70  16 

22 

71  50 

25 

73  11 

87 

82  23 

66  55 

28 

69  35 

31 

71  43 

34 

73  28 

36 

74  56 

38 

76  10 

40 

77  14 

88 

43 

74  8 

45 

76  3 

47 

77  34 

48 

78  48 

49 

79  49 

50 

80  41 

51 

81  24 

89 

56 

81  55 

56 

82  55 

57 

83  43 

57 

84  21 

57 

84  52 

58 

85  18 

58 

85  41 

90 

83  0 

90  0 

82  0 

90  0 

81  0 

90  0 

80  0 

90  0 

79  0 

90  0 

78  0 

90  0 

77  0 

90  0 

296 


Table  13.    Kelvin's  Sumner  Line  Table 


b. 

a  = 

--  14° 

a  = 

--  15° 

a  = 

=  16° 

a  = 

=  17° 

a  = 

--  18° 

a  = 

--  19° 

a  = 

-  30° 

K 

Q 

K 

Q 

K 

Q 

K 

Q 

K 

Q 

K 

Q 

K 

i 

c 

o 

o 

o   / 

o   / 

o 

0   / 

o 

o   / 

o 

o   / 

o 

o   / 

o   / 

o   / 

0 

0  0 

14  0 

0  0 

15  0 

0  0 

16  0 

0  0 

17  0 

0  0 

18  0 

0  0 

19  0 

0  0 

20  0 

1 

58 

0 

58 

0 

58 

0 

57 

0 

57 

0 

57 

0 

56 

0 

2 

1  56 

0 

1  56 

1 

1  55 

1 

1  55 

1 

1  54 

1 

154 

1 

1  53 

1 

3 

2  55 

1 

2  54 

1 

2  53 

1 

2  52 

1 

2  51 

1 

2  50 

2 

2  49 

2 

4 

3  53 

2 

3  52 

2 

3  51 

2 

3  49 

2 

3  48 

2 

3  47 

3 

3  46 

3 

5 

4  51 

3 

4  50 

3 

4  48 

3 

4  47 

'4 

4  45 

4 

4  44 

4 

4  42 

4 

G 

5  49 

4 

5  48 

5 

5  46 

5 

5  44 

5 

5  42 

6 

5  40 

6 

5  38 

6 

7 

6  47 

6 

6  46 

7 

6  44 

7 

6  42 

7 

6  39 

8 

6  37 

8 

6  35 

8 

8 

7  46 

8 

7  44 

9 

7  41 

9 

7  39 

9 

7  36 

10 

7  34 

10 

7  31 

11 

9 

8  44 

10 

8  41 

11 

8  39 

11 

8  36 

12 

8  33 

13 

8  30 

13 

8  27 

14 

10 

9  42 

12 

9  39 

13 

9  37 

14 

9  34 

15 

9  30 

16 

9  27 

16 

9  23 

17 

11 

10  40 

15 

10  37 

16 

10  34 

17 

10  31 

18 

10  27 

19 

10  24 

20 

10  20 

21 

12 

11  38 

18 

11  35 

19 

11  32 

20 

11  28 

21 

11  24 

23 

11  20 

24 

11  16 

25 

13 

12  36 

21 

12  33 

22 

12  29 

24 

12  25 

25 

12  21 

27 

12  17 

28 

12  12 

29 

14 

13  35 

25 

13  31 

26 

13  27 

28 

13  23 

29 

13  18 

31 

13  13 

32 

13  8 

34 

15 

14  33 

28 

14  29 

30 

14  24 

32 

14  20 

34 

14  15 

36 

14  10 

37 

14  5 

39 

16 

15  31 

32 

15  26 

35 

15  22 

37 

15  17 

39 

15  12 

41 

15  6 

42 

15  1 

44 

17 

16  29 

37 

16  24 

39 

16  19 

42 

16  14 

44 

16  9 

46 

16  3 

48 

57 

50 

18 

17  27 

41 

17  22 

44 

17  17 

47 

17  11 

49 

17  6 

52 

59 

54 

16  53 

56 

19 

18  25 

46 

18  20 

49 

18  14 

52 

18  8 

55 

18  2 

58 

17  56 

20  1 

17  49 

21  3 

20 

19  23 

52 

19  17 

55 

19  12 

58 

19  5 

18  1 

59 

19  4 

18  52 

8 

18  45 

10 

21 

20  21 

57 

20  15 

16  1 

20  9 

17  4 

20  2 

8 

19  56 

11 

19  48 

15 

19  41 

18 

22 

21  19 

15  3 

21  13 

7 

21  6 

11 

59 

15 

20  52 

19 

20  45 

22 

20  37 

26 

23 

22  17 

9 

22  10 

14 

22  4 

18 

21  56 

22 

21  49 

27 

21  41 

30 

21  32 

34 

24 

23  15 

16 

23  8 

21 

23  1 

26 

22  53 

30 

22  45 

35 

22  37 

39 

22  28 

43 

25 

24  13 

23 

24  6 

28 

58 

34 

23  50 

38 

23  42 

43 

23  33 

48 

23  24 

53 

26 

25  10 

30 

25  3 

36 

24  55 

42 

24  47 

47 

24  38 

52 

24  29 

58 

24  20 

22  3 

27 

26  8 

38 

26  1 

44 

25  52 

50 

25  44 

56 

25  35 

20  2 

25  25 

21  8 

25  15 

13 

28 

27  6 

46 

58 

53 

26  50 

59 

26  41 

19  6 

26  31 

12 

26  21 

18 

26  11 

24 

29 

28  4 

55 

27  55 

17  2 

27  47 

18  9 

27  37 

16 

27  27 

23 

27  17 

29 

27  6 

36 

30 

29  1 

16  4 

28  53 

12 

28  44 

19 

28  34 

27 

28  24 

34 

28  13 

41 

28  2 

48 

31 

59 

13 

29  50 

22 

29  41 

30 

29  30 

38 

29  20 

46 

29  9 

53 

57 

23  1 

32 

30  57 

23 

30  47 

32 

30  37 

41 

30  27 

50 

30  16 

58 

30  4 

22  6 

29  52 

14 

33 

31  54 

33 

31  44 

43 

31  34 

53 

31  23 

20  2 

31  12 

21  11 

31  0 

19 

30  47 

28 

34 

32  52 

44 

32  42 

55 

32  31 

19  5 

32  20 

15 

32  8 

24 

55 

33 

31  42 

42 

35 

33  49 

56 

33  39 

18  7 

33  28 

18 

33  16 

28 

33  4 

38 

32  51 

48 

32  37 

57 

36 

34  46 

17  8 

34  36 

20 

34  24 

31 

34  12 

42 

59 

53 

33  46 

23  3 

33  32 

24  13 

37 

35  44 

20 

35  33 

33 

35  21 

45 

35  8 

57 

34  55 

22  8 

34  41 

19 

34  26 

30 

38 

36  41 

33 

36  29 

47 

36  17 

20  0 

36  4 

21  12 

35  50 

24 

35  36 

36 

35  21 

48 

39 

37  38 

47 

37  26 

19  1 

37  13 

15 

37  0 

28 

36  46 

41 

36  31 

54 

36  15 

25  6 

40 

38  35 

18  2 

38  23 

17 

38  10 

31 

56 

45 

37  41 

59 

37  26 

24  12 

37  9 

25 

41 

39  32 

17 

39  19 

33 

39  6 

48 

38  52 

22  3 

38  36 

23  18 

38  20 

32 

38  3 

45 

42 

40  29 

33 

40  16 

50 

40  2 

21  6 

39  47 

22 

39  31 

37 

39  15 

52 

57 

26  6 

43 

41  26 

•  50 

41  12 

20  7 

58 

25 

40  42 

41 

40  26 

57 

40  9 

25  13 

39  51 

27 

44 

42  23 

19  7 

42  9 

26 

4154 

44 

41  38 

23  2 

41  21 

24  18 

41  3 

35 

40  45 

50 

45 

43  19 

25 

43  5 

45 

42  49 

22  4 

42  33 

23 

42  16 

41 

57 

58 

4139 

27  14 

Table  13.    Kelvin's  Sumner  Line  Table 


297 


b 

a  =  14° 

a  =  15°    a  =  16° 

a  =  17° 

a  =  18° 

a  =  19° 

a  =  20° 

K 

Q 

K 

Q 

K 

Q 

K 

Q 

K 

Q 

K 

Q 

K 

Q 

0 

o    / 

o    / 

0     / 

o    / 

o   / 

o   / 

0    / 

o    / 

o   / 

0    / 

o   / 

o   / 

o    / 

o   / 

45 

43  19 

19  25 

43  5 

20  45 

42  49 

22  4 

42  33 

23  23 

42  16 

24  41 

41  57 

25  58 

41  39 

27  14 

46 

44  16 

45 

44  1 

21  6 

43  45 

26 

43  28 

45 

43  10 

25  4 

42  51 

26  22 

42  32 

39 

47 

45  12 

20  5 

57 

27 

44  40 

48 

44  23 

24  9 

44  4 

28 

43  45 

47 

43  25 

28  5 

48 

46  8 

26 

45  53 

49 

45  35 

23  12 

45  18 

33 

58 

54 

44  39 

27  14 

44  18 

33 

49 

47  4 

48 

46  48 

22  13 

46  30 

37 

46  12 

59 

45  52 

26  21 

45  32 

42 

45  10 

29  1 

50 

48  0 

21  12 

47  44 

38 

47  25 

24  3 

47  6 

25  26 

46  46 

49 

46  25 

28  11 

46  2 

31 

51 

56 

37 

48  39 

23  4 

48  20 

30 

48  0 

55 

47  39 

27  18 

47  18 

41 

54 

30  3 

52 

49  52 

22  3 

49  34 

31 

49  15 

59 

54 

26  25 

48  32 

49 

48  10 

29  13 

47  46 

36 

53 

50  48 

30 

50  29 

24  0 

50  9 

25  29 

49  48 

56 

49  25 

28  22 

49  2 

47 

48  38 

31  10 

54 

51  43 

59 

51  24 

30 

51  3 

26  0 

50  41 

27  29 

50  18 

56 

54 

30  22 

49  29 

46 

55 

52  38 

23  30 

52  18 

25  2 

57 

34 

51  34 

28  4 

51  10 

29  32 

50  46 

59 

50  20 

32  24 

5G 

53  33 

24  2 

53  12 

36 

52  50 

27  9 

52  27 

40 

52  2 

30  10 

51  37 

31  37 

51  10 

33  3 

57 

54  28 

36 

54  6 

26  12 

53  43 

46 

53  20 

29  18 

54 

49 

52  28 

32  18 

52  0 

45 

58 

55  22 

25  12 

55  0 

49 

54  36 

28  25 

54  12 

59 

53  46 

31  31 

53  18 

33  1 

50 

34  29 

59 

56  16 

50 

53 

27  29 

55  29 

29  6 

55  4 

30  42 

54  37 

32  15 

54  8 

46 

53  39 

35  15 

60 

57  10 

26  30 

56  46 

28  11 

56  21 

50 

55 

31  27 

55  27 

33  1 

58 

34  33 

54  28 

36  3 

61 

58  4 

27  13 

57  39 

56 

57  13 

30  36 

56  46 

32  14 

56  17 

50 

55  47 

35  23 

55  16 

54 

62 

57 

59 

58  31 

29  43 

58  4 

31  25 

57  36 

33  4 

57  7 

34  41 

56  36 

36  16 

56  4 

37  47 

63 

59  50 

28  47 

59  23 

30  33 

55 

32  17 

58  26 

57 

56 

35  35 

57  24 

37  11 

51 

38  43 

64 

60  42 

29  38 

60  15 

31  26 

59  46 

33  11 

59  16 

34  53 

58  44 

36  33 

58  11 

38  9 

57  38 

39  42 

65 

6134 

30  32 

61  6 

32  23 

60  36 

34  9 

60  5 

35  53 

59  32 

37  33 

58 

39  10 

58  24 

40  44 

66 

62  25 

31  31 

56 

33  23 

61  25 

35  11 

53 

36  56 

60  19 

38  37 

59  44 

40  15 

59  9 

41  49 

67 

63  16 

32  33 

62  46 

34  27 

62  14 

36  17 

61  41 

38  3 

61  6 

39  45 

60  30 

41  23 

53 

42  58 

68 

64  7 

33  39 

63  35 

35  35 

63  2 

37  26 

62  28 

39  13 

52 

40  56 

61  15 

42  35 

60  36 

44  11 

69 

56 

34  50 

64  23 

36  47 

49 

38  40 

63  14 

40  28 

62  37 

42  12 

58 

43  51 

61  19 

45  27 

70 

65  45 

36  5 

65  11 

38  4 

64  36 

39  58 

59 

4148 

63  21 

43  32 

62  41 

45  11 

62  1 

46  47 

71 

66  33 

37  27 

58 

39  27 

65  21 

41  22 

64  43 

43  12 

64  4 

44  56 

63  23 

46  36 

41 

48  11 

72 

67  20 

38  54 

66  44 

40  56 

66  6 

42  52 

65  26 

44  42 

45 

46  26 

64  4 

48  6 

63  21 

49  40 

73 

68  7 

40  27 

67  29 

42  30 

49 

44  27 

66  8 

46  17 

65  26 

48  1 

43 

49  40 

59 

51  14 

74 

52 

42  8 

68  12 

44  11 

67  31 

46  8 

49 

47  58 

66  6 

49  42 

65  21 

51  19 

64  36 

52  52 

75 

69  36 

43  56 

55 

45  59 

68  12 

47  56 

67  29 

49  45 

44 

51  28 

58 

53  4 

65  11 

54  35 

76 

70  18 

45  52 

69  36 

47  55 

52 

49  51 

68  7 

51  39 

67  20 

53  20 

66  33 

54  55 

45 

56  23 

77 

59 

47  57 

70  15 

49  59 

69  30 

51  53 

43 

53  39 

55 

55  18 

67  7 

56  51 

66  18 

58  17 

78 

71  38 

50  11 

53 

52  12 

70  6 

54  3 

69  18 

55  47 

68  29 

57  23 

39 

58  53 

48 

60  16 

79 

72  16 

52  35 

7128 

54  33 

40 

56  21 

50 

58  2 

69  0 

59  35 

68  9 

61  0 

67  17 

62  20 

80 

51 

55  9 

72  2 

57  3 

71  12 

58  48 

70  21 

60  24 

29 

61  53 

37 

63  14 

44 

64  30 

81 

73  24 

57  54 

34 

59  43 

42 

61  23 

50 

62  54 

56 

64  18 

69  3 

65  34 

68  9 

66  45 

82 

55 

60  50 

73  3 

62  33 

72  9 

64  7 

71  16 

65  31 

70  21 

66  49 

27 

68  0 

31 

69  5 

83 

74  23 

63  57 

29 

65  32 

34 

66  59 

39 

68  16 

44 

69  26 

48 

70  31 

51 

7129 

84 

48 

67  15 

52 

68  41 

56 

69  59 

72  0 

71  7 

71  3 

72  10 

70  7 

73  7 

69  9 

73  59 

85 

75  9 

70  44 

74  12 

7159 

73  15 

73  6 

18 

74  5 

20 

74  59 

23 

75  48 

25 

76  32 

86 

27 

74  22 

29 

75  25 

31 

76  20 

33 

77  9 

35 

77  53 

36 

78  33 

37 

79  9 

87 

41 

78  9 

43 

78  57 

44 

79  39 

45 

80  17 

46 

80  51 

46 

81  22 

47 

81  49 

88 

52 

82  2 

52 

82  35 

53 

83  4 

53 

83  29 

54 

83  52 

54 

84  13 

54 

84  31 

89 

58 

86  0 

58 

86  16 

58 

86  31 

58 

86  44 

58 

86  55 

58 

87  6 

59 

87  15 

90 

76  0 

90  0 

75  0 

90  0 

74  0 

90  0 

73  0 

90  0 

72  0 

90  0 

71  0 

90  0 

70  0 

90  0 

298 


Table  13.    Kelvin's  Sumner  Line  Table 


b 

a  = 

21° 

a  = 

22° 

a  = 

33° 

a  = 

34° 

a  = 

25° 

a  = 

26° 

a  = 

27° 

K 

Q 

o   / 

K 

Q 

K 

Q 

K 

Q 

K 

Q 

K 

Q 

K 

Q 

o 

o   / 

o     / 

o    / 

o   / 

o   / 

o    / 

o    / 

°     1 

o   / 

o   / 

0    / 

o    / 

0    / 

0 

0  0 

21  0 

0  0 

22  0 

0  0 

23  0 

0  0 

24  0 

0  0 

25  0 

0  0 

26  0 

0  0 

27  0 

1 

56 

0 

56 

0 

55 

0 

55 

0 

54 

0 

54 

0 

53 

0 

2 

1  52 

1 

1  51 

1 

1  51 

1 

1  50 

1 

1  49 

1 

1  48 

1 

1  47 

1 

3 

2  48 

2 

2  47 

2 

2  46 

2 

2  44 

2 

2  43 

2 

2  42 

2 

2  40 

2 

4 

3  44 

3 

3  42 

3 

3  41 

3 

3  39 

3 

3  38 

3 

3  36 

3 

3  34 

3 

5 

4  40 

4 

4  38 

5 

4  36 

5 

4  34 

5 

4  32 

5 

4  30 

5 

4  27 

5 

6 

5  36 

6 

5  34 

7 

5  31 

7 

5  29 

7 

5  26 

7 

5  23 

7 

5  21 

8 

7 

6  32 

9 

6  29 

9 

6  26 

9 

6  24 

10 

6  20 

10 

6  17 

10 

6  14 

11 

8 

7  28 

11 

7  25 

12 

7  22 

12 

7  18 

13 

7  15 

13 

7  11 

13 

7  7 

14 

9 

8  24 

14 

8  20 

15 

8  17 

15 

8  13 

16 

8  9 

16 

8  5 

17 

8  1 

17 

10 

9  20 

18 

9  16 

18 

9  12 

19 

9  8 

20 

9  3 

20 

59 

21 

54 

21 

11 

10  16 

22 

10  11 

22 

10  7 

23 

10  2 

24 

57 

24 

9  53 

25 

9  47 

26 

12 

11  12 

26 

11  7 

26 

11  2 

27 

57 

28 

10  52 

29 

10  46 

30 

10  41 

31 

13 

12  7 

30 

12  2 

31 

57 

32 

11  52 

33 

11  46 

34 

11  40 

35 

11  34 

36 

14 

13  3 

35 

58 

36 

12  52 

38 

12  46 

39 

12  40 

40 

12  34 

41 

12  27 

42 

15 

59 

40 

13  53 

42 

13  47 

43 

13  41 

45 

13  34 

46 

13  27 

47 

13  20 

49 

16 

14  55 

46 

14  48 

48 

14  42 

49 

14  35 

51 

14  28 

53 

14  21 

54 

14  13 

56 

17 

15  50 

52 

15  44 

54 

15  37 

56 

15  29 

58 

15  22 

26  0 

15  14 

27  1 

15  6 

28  3 

18 

16  46 

59 

16  39 

23  1 

16  32 

24  3 

16  24 

25  5 

16  16 

7 

16  8 

9 

59 

11 

19 

17  42 

22  6 

17  34 

8 

17  26 

11 

17  18 

13 

17  10 

15 

17  1 

17 

16  52 

19 

20 

18  37 

13 

18  29 

16 

18  21 

19 

18  12 

21 

18  3 

23 

54 

26 

17  45 

28 

21 

19  33 

21 

19  24 

24 

19  16 

27 

19  7 

30 

57 

32 

18  47 

35 

18  37 

37 

22 

20  28 

29 

20  19 

33 

20  10 

36 

20  1 

39 

19  51 

42 

19  40 

45 

19  30 

47 

23 

21  24 

38 

21  14 

42 

21  5 

45 

55 

49 

20  44 

52 

20  33 

55 

20  22 

58 

24 

22  19 

47 

22  9 

52 

59 

55 

2149 

59 

21  38 

27  3 

21  26 

28  6 

21  15 

29  9 

25 

23  14 

57 

23  4 

24  2 

22  54 

25  6 

22  43 

26  10 

22  31 

14 

22  19 

17 

22  7 

21 

26 

24  9 

23  8 

59 

12 

23  48 

17 

23  37 

21 

23  25 

25 

23  12 

29 

23  0 

33 

27 

25  4 

19 

24  54 

23 

24  42 

.  28 

24  30 

33 

24  18 

37 

24  5 

42 

52 

46 

28 

59 

30 

25  48 

35 

25  36 

40 

25  24 

46 

25  11 

50 

57 

55 

24  44 

59 

29 

26  54 

42 

26  43 

48 

26  30 

53 

26  17 

59 

26  4 

28  4 

25  50 

29  9 

25  36 

30  13 

30 

27  49 

54 

27  37 

25  1 

27  24 

26  7 

27  11 

27  13 

57 

18 

26  42 

23 

26  27 

28 

31 

28  44 

24  7 

28  32 

14 

28  18 

21 

28  4 

27 

27  50 

33 

27  34 

38 

27  19 

44 

32 

29  39 

21 

29  26 

28 

29  12 

35 

57 

42 

28  42 

48 

28  26 

54 

28  11 

31  0 

33 

30  34 

36 

30  20 

43 

30  5 

51 

29  50 

58 

29  35 

29  4 

29  18 

30  11 

29  2 

17 

34 

31  28 

51 

31  14 

59 

59 

27  7 

30  43 

28  14 

30  27 

21 

30  10 

28 

53 

35 

35 

32  23 

25  7 

32  8 

26  15 

31  52 

24 

31  36 

31 

31  19 

39 

31  2 

46 

30  44 

53 

36 

33  17 

23 

33  1 

32 

32  45 

41 

32  29 

49 

32  11 

58 

53 

31  5 

31  35 

32  12 

37 

34  11 

40 

55 

50 

33  38 

59 

33  21 

29  8 

33  3 

30  17 

32  45 

25 

32  26 

32 

38 

35  5 

58 

34  48 

27  9 

34  31 

28  18 

34  13 

28 

55 

37 

33  36 

45 

33  16 

53 

39 

59 

26  17 

35  42 

28 

35  24 

38 

35  5 

49 

34  47 

58 

34  27 

32  7 

34  6 

33  15 

40 

36  53 

37 

36  35 

48 

36  17 

59 

57 

30  10 

35  38 

31  20 

35  18 

29 

56 

38 

41 

37  46 

58 

37  28 

28  10 

37  9 

29  21 

36  49 

32 

36  29 

43 

36  8 

52 

35  46 

34  1 

42 

38  40 

27  19 

38  21 

32 

38  1 

44 

37  41 

56 

37  20 

32  7 

58 

33  17 

36  36 

26 

43 

39  33 

42 

39  13 

55 

53 

30  8 

38  32 

31  20 

38  11 

31 

37  48 

42 

37  25 

52 

44 

40  26 

28  5 

40  6 

29  19 

39  45 

33 

39  23 

45 

39  1 

57 

38  38 

34  8 

38  14 

35  19 

45 

41  19 

30 

58 

44 

40  37 

59 

40  14 

32  12 

51 

33  24 

39  28 

36 

39  3 

47 

Table  13.    Kelvin's  Sumuer  Line  Table 


299 


b 

a  =  21° 

a  =  22° 

a  =  23° 

a  =  24° 

a  =  25° 

a  =  26° 

a  =  37° 

K 

Q 

K 

Q 

K 

Q 

K 

Q 

K 

Q 

K 

Q 

K 

Q 

o 

o   / 

o   / 

o   / 

o   / 

o   / 

o   / 

o   / 

o   / 

o   / 

0    / 

o   / 

o   / 

o   / 

0    / 

45 

41  19 

28  30 

40  58 

29  44 

40  37 

30  59 

40  14 

32  12 

39  51 

33  24 

39  28 

34  36 

39  3 

35  47 

46 

42  11 

55 

41  50 

30  11 

41  28 

31  26 

41  5 

39 

40  41 

52 

40  17 

35  4 

52 

36  16 

47 

43  4 

29  22 

42  42 

39 

42  19 

54 

55 

33  8 

41  31 

34  22 

41  6 

34 

40  40 

46 

48 

56 

50 

43  33 

31  8 

43  10 

32  23 

42  45 

38 

42  20 

52 

55 

36  5 

41  28 

37  17 

49 

44  48 

30  20 

44  24 

38 

44  0 

54 

43  35 

34  10 

43  9 

35  24 

42  43 

37 

42  15 

50 

50 

45  40 

51 

45  15 

32  9 

50 

33  26 

44  25 

43 

58 

57 

43  31 

37  11 

43  2 

38  24 

51 

46  31 

31  23 

46  6 

42 

45  40 

34  0 

45  14 

35  17 

44  47 

36  32 

44  18 

46 

49 

39  0 

52 

47  22 

57 

56 

33  17 

46  30 

35 

46  3 

53 

45  35 

37  8 

45  5 

38  23 

44  36 

37 

53 

48  13 

32  32 

47  46 

53 

47  19 

35  12 

51 

36  30 

46  22 

46 

52 

39  1 

45  22 

40  15 

54 

49  3 

33  9 

48  36 

34  30 

48  8 

50 

47  39 

37  9 

47  9 

38  26 

46  39 

41 

46  7 

55 

55 

53 

48 

49  25 

35  10 

56 

36  30 

48  27 

49 

56 

39  7 

47  25 

40  23 

52 

41  37 

56 

50  43 

34  28 

50  14 

51 

49  44 

37  12 

49  14 

38  32 

48  42 

49 

48  10 

41  6 

47  37 

42  20 

57 

51  32 

35  11 

51  2 

36  34 

50  32 

56 

50  1 

39  16 

49  28 

40  34 

55 

51 

48  21 

43  5 

58 

52  21 

55 

50 

37  19 

51  19 

38  42 

47 

40  2 

50  14 

41  21 

49  40 

42  38 

49  5 

52 

59 

53  9 

36  42 

52  38 

38  7 

52  6 

39  30 

51  33 

50 

59 

42  9 

50  24 

43  26 

48 

44  41 

60 

57 

37  31 

53  25 

56 

52 

40  20 

52  18 

41  41 

51  43 

43  0 

51  7 

44  17 

50  30 

45  32 

61 

54  44 

38  22 

54  11 

39  48 

53  37 

41  12 

53  2 

42  34 

52  26 

53 

49 

45  10 

51  12 

46  25 

62 

55  31 

39  16 

57 

40  43 

54  22 

42  7 

46 

43  29 

53  9 

44  48 

52  31 

46  6 

53 

47  21 

63 

56  17 

40  13 

55  42 

41  40 

55  6 

43  5 

54  29 

44  27 

51 

45  46 

53  13 

47  3 

52  33 

48  18 

64 

57  3 

41  13 

56  27 

42  40 

49 

44  5 

55  11 

45  27 

54  33 

46  46 

53 

48  3 

53  13 

49  18 

65 

48 

42  15 

57  11 

43  43 

56  32 

45  8 

53 

46  30 

55  13 

47  49 

54  33 

49  5 

51 

50  20 

66 

58  32 

43  21 

54 

44  49 

57  14 

46  13 

56  34 

47  35 

53 

48  54 

55  12 

50  10 

54  29 

51  24 

67 

59  15 

44  30 

58  36 

45  58 

55 

47  22 

57  14 

48  44 

56  32 

50  2 

50 

51  18 

55  6 

52  31 

68 

57 

45  42 

59  17 

47  10 

58  35 

48  34 

53 

49  55 

57  10 

51  13 

56  27 

52  28 

42 

53  41 

69 

60  39 

46  58 

57 

48  26 

59  15 

49  50 

58  31 

51  10 

47 

52  27 

57  3 

53  42 

56  17 

54  53 

70 

61  19 

48  18 

60  36 

49  45 

53 

51  8 

59  8 

52  28 

58  23 

53  44 

38 

54  58 

51 

56  8 

71 

58 

49  42 

61  14 

51  8 

60  30 

52  31 

44 

53  49 

58 

55  5 

58  12 

56  17 

57  24 

57  25 

72 

62  36 

51  10 

51 

52  35 

61  6 

53  57 

60  19 

55  14 

59  32 

56  28 

44 

57  39 

56 

58  46 

73 

63  13 

52  42 

62  27 

54  7 

41 

55  27 

53 

56  42 

60  5 

57  55 

59  16 

59  4 

58  26 

60  9 

74 

49 

54  19 

63  2 

55  42 

62  14 

57  0 

61  25 

58  14 

36 

59  25 

46 

60  32 

55 

61  35 

75 

64  23 

56  1 

35 

57  21 

46 

58  38 

56 

59  50 

61  6 

60  58 

60  15 

62  3 

59  23 

63  4 

76 

56 

57  47 

64  7 

59  5 

63  16 

60  19 

62  26 

61  29 

34 

62  35 

42 

63  37 

50 

64  36 

77 

65  27 

59  38 

37 

60  53 

45 

62  5 

54 

63  12 

62  1 

64  15 

61  8 

65  14 

00  15 

66  11 

78 

57 

61  34 

65  5 

62  46 

64  13 

63  54 

63  20 

64  58 

26 

65  58 

32 

66  55 

38 

67  48 

79 

66  25 

63  34 

32 

64  43 

38 

65  48 

44 

66  48 

50 

67  45 

55 

68  38 

61  0 

69  28 

80 

50 

65  40 

56 

66  44 

65  2 

67  45 

64  7 

68  42 

63  12 

69  35 

62  16 

70  24 

20 

71  11 

81 

67  14 

67  50 

66  19 

68  50 

23 

69  46 

28 

70  39 

32 

71  27 

35 

72  13 

39 

72  56 

82 

36 

70  4 

40 

71  0 

43 

71  51 

47 

72  39 

50 

73  23 

53 

74  4 

56 

74  43 

83 

55 

72  23 

58 

73  13 

66  1 

73  59 

65  3 

74  42 

64  6 

75  21 

63  8 

75  58 

62  10 

76  33 

84 

68  12 

74  46 

67  14 

75  30 

16 

76  10 

18 

76  47 

20 

77  22 

22 

77  54 

23 

78  24 

85 

26 

77  12 

28 

77  50 

29 

78  24 

31 

78  55 

32 

79  25 

33 

79  52 

35 

80  18 

86 

38 

79  42 

39 

80  12 

40 

80  40 

41 

81  6 

42 

81  30 

43 

81  52 

44 

82  12 

87 

48 

82  14 

48 

82  37 

49 

82  58 

49 

83  18 

50 

83  36 

50 

83  53 

51 

84  8 

88 

55 

84  48 

55 

85  4 

55 

85  18 

55 

85  31 

56 

85  43 

56 

85  55 

56 

86  5 

89 

59 

87  24 

59 

87  32 

59 

87  39 

59 

87  45 

59 

87  51 

59 

87  57 

59 

88  2 

90 

69  0 

90  0 

68  0 

90  0 

67  0 

90  0 

66  0 

90  0 

65  0 

90  0 

64  0 

90  0 

63  0 

90  0 

300 


Table  13.    Kelvin's  Sumner  Line  Table 


b 

o 

a  = 

28° 

a  = 

39° 

a  = 

30° 

a  =  31° 

a  =  33° 

a  =  33° 

a  =  34° 

K 

Q 

K 

Q 

K 

Q 

K 

Q 

K 

Q 

K 

Q 

K 

Q 

o   / 

0    / 

o   / 

o    / 

o    / 

o    / 

o   / 

o   / 

o   / 

o    / 

O    / 

o   / 

0    / 

o   / 

0 

0  0 

28  0 

0  0 

29  0 

0  0 

30  0 

0  0 

31  0 

0  0 

32  0 

0  0 

33  0 

0  0  34  0  1 

1 

53 

0 

52 

0 

52 

0 

51 

0 

51 

0 

50 

0 

50 

0 

2 

1  46 

1 

145 

1 

1  44 

1 

143 

1 

142 

1 

141 

1 

1  39 

1 

3 

2  39 

2 

2  37 

2 

2  36 

2 

2  34 

2 

2  33 

2 

2  31 

2 

2  29 

2 

4 

3  32 

3 

3  30 

4 

3  28 

4 

3  26 

4 

3  23 

4 

3  21 

4 

3  19 

4 

5 

4  25 

5 

4  22 

6 

4  20 

6 

4  17 

6 

4  14 

6 

4  12 

6 

4  9 

6 

6 

5  18 

8 

5  15 

8 

5  12 

8 

5  8 

8 

5  5 

8 

5  2 

.   9 

58 

9 

7 

6  11 

11 

6  7 

11 

6  4 

11 

6  0 

11 

56 

11 

52 

12 

5  48 

12 

8 

7  4 

14 

59 

14 

55 

15 

51 

15 

6  47 

15 

6  42 

15 

6  38 

16 

9 

56 

18 

7  52 

18 

7  47 

19 

7  43 

19 

7  37 

19 

7  32 

19 

7  27 

20 

10 

8  49 

22 

8  44 

22 

8  39 

23 

8  34 

23 

8  28 

24 

8  22 

24 

8  17 

25 

11 

9  42 

27 

9  36 

27 

9  31 

28 

9  25 

28 

9  19 

29 

9  12 

29 

9  6 

30 

12 

10  35 

32 

10  29 

32 

10  22 

33 

10  16 

34 

10  9 

34 

10  2 

35 

56 

35 

13 

11  27 

37 

11  21 

38 

11  14 

39 

11  7 

40 

11  0 

40 

52 

41 

10  45 

41 

14 

12  20 

43 

12  13 

44 

12  6 

45 

58 

46 

50 

47 

11  42 

48 

11  34 

48 

15 

13  13 

50 

13  5 

51 

57 

52 

12  49 

53 

12  41 

54 

12  32 

55 

12  23 

56 

16 

14  5 

57 

57 

58 

13  49 

59 

13  40 

32  1 

13  31 

33  2 

13  22 

34  3 

13  13 

35  4 

17 

58 

29  4 

14  49 

30  6 

14  40 

31  7 

14  31 

9 

14  21 

10 

14  12 

11 

14  2 

12 

18 

15  50 

12 

15  41 

14 

15  31 

16 

15  22 

17 

15  12 

18 

15  1 

20 

51 

21 

19 

16  42 

21 

16  33 

23 

16  23 

25 

16  12 

26 

16  2 

27 

51 

29 

15  40 

30 

20 

17  35 

30 

17  24 

32 

17  14 

34 

17  3 

36 

52 

37 

16  40 

39 

16  28 

40 

21 

18  27 

40 

18  16 

42 

18  5 

44 

53 

46 

17  42 

48 

17  29 

49 

17  17 

51 

22 

19  19 

50 

19  8 

52 

56 

55 

18  44 

57 

18  31 

59 

18  19 

35  0 

18  6 

36  2 

23 

20  11 

30  1 

59 

31  3 

19  47 

32  6 

19  34 

33  8 

19  21 

34  10 

19  8 

12 

54 

14 

24 

21  3 

12 

20  50 

15 

20  37 

18 

20  24 

20 

20  11 

22 

57 

24 

19  42 

26 

25 

55 

24 

21  41 

27 

21  28 

30 

21  14 

33 

21  0 

35 

20  46 

37 

20  30 

39 

26 

22  46 

37 

22  32 

40 

22  19 

43 

22  4 

46 

49 

48 

21  34 

51 

21  18 

53 

27 

23  38 

50 

23  23 

53 

23  9 

57 

54 

34  0 

22  38 

35  2 

22  23 

36  5 

22  6 

37  8 

28 

24  29 

31  3 

24  14 

32  7 

59 

33  11 

23  44 

14 

23  27 

17 

23  11 

20 

54 

23 

29 

25  21 

18 

25  5 

22 

24  49 

20 

24  33 

29 

24  16 

33 

59 

36 

23  42 

38 

30 

26  12 

33 

56 

37 

25  39 

41 

25  23 

45 

25  5 

49 

24  47 

52 

24  29 

55 

31 

27  3 

49 

26  47 

53 

26  29 

58 

26  12 

35  2 

54 

36  6 

25  35 

37  9 

25  16 

38  12 

32 

54 

32  5 

27  37 

33  10 

27  19 

34  15 

27  1 

19 

26  42 

23 

26  23 

27 

26  3 

30 

33 

28  45 

22 

28  27 

28 

28  9 

33 

50 

37 

27  30 

41 

27  11 

45 

50 

49 

34 

29  35 

40 

29  17 

46 

58 

51 

28  39 

56 

28  18 

37  0 

58 

38  4 

27  37 

39  8 

35 

30  26 

59 

30  7 

34  5 

29  47 

35  11 

29  27 

36  16 

29  6 

20 

28  45 

24 

28  24 

28 

36 

31  16 

33  19 

56 

25 

30  36 

31 

30  15 

36 

54 

41 

29  32 

45 

29  10 

49 

37 

32  6 

39 

31  46 

46 

31  25 

52 

31  3 

57 

30  41 

38  2 

30  19 

39  7 

56 

40  11 

38 

56 

34  1 

32  35 

35  7 

32  13 

36  14 

51 

37  20 

31  28 

25 

31  5 

30 

30  42 

34 

39 

33  46 

23 

33  24 

30 

33  1 

37 

32  39 

43 

32  15 

48 

51 

53 

31  27 

57 

40 

34  35 

46 

34  13 

53 

49 

37  0 

33  26 

38  7 

33  2 

39  12 

32  37 

40  17 

32  12 

41  22 

41 

35  24 

35  10 

35  1 

36  18 

34  37 

25 

34  13 

31 

48 

37 

33  23 

43 

57 

47 

42 

36  13 

35 

49 

43 

35  25 

51 

35  0 

57 

34  34 

40  4 

34  8 

41  9 

33  42 

42  14 

43 

37  2 

36  1 

36  37 

37  10 

36  12 

38  17 

46 

39  24 

35  20 

31 

53 

36 

34  26 

41 

44 

50 

28 

37  25 

37 

59 

45 

36  33 

52 

36  6 

59 

35  38 

42  5 

35  10 

43  10 

45 

38  38 

57 

38  12 

38  5 

37  46 

39  14 

37  19 

40  21 

51 

41  28 

36  22 

34 

53 

39 

Table  13.    Kelvin's  Sumner  Line  Table 


301 


b 

a  =  28° 

a  =  29° 

a  =  30° 

a  =  31° 

a  =  32° 

a  =  33° 

a  =  34° 

K 

Q 

K 

Q 

K 

Q 

K 

Q 

K 

Q 

K 

Q 

K 

Q 

o 

o   / 

o   / 

O    / 

0    / 

o    / 

o   / 

o   / 

o   / 

o   / 

o   / 

o   / 

O    1 

o   / 

0    / 

45 

38  38 

36  57 

38  12 

38  5 

37  46 

39  14 

37  19 

40  21 

36  51 

41  28 

36  22 

42  34 

35  53 

43  39 

46 

39  26 

37  26 

59 

35 

38  32 

44 

38  4 

52 

37  36 

58 

37  6 

43  4 

36  36 

44  9 

47 

40  13 

56 

39  46 

39  6 

39  18 

40  15 

49 

41  23 

38  20 

42  30 

50 

36 

37  19 

41 

48 

41  0 

38  28 

40  32 

38 

40  4 

47 

39  34 

55 

39  4 

43  2 

38  33 

44  9 

38  2 

45  14 

49 

47 

39  1 

41  18 

40^11 

49 

41  21 

40  19 

42  29 

48 

36 

39  16 

43 

44 

48 

50 

42  34 

36 

42  4 

46 

41  34 

56 

41  3 

43  4 

40  31 

44  11 

59 

45  18 

39  26 

46  23 

51 

43  20 

40  12 

49 

41  22 

42  18 

42  32 

46 

40 

41  14 

48 

40  41 

54 

40  7 

59 

52 

44  5 

49 

43  34 

42  0 

43  2 

43  10 

42  29 

44  18 

56 

45  26 

41  22 

46  32 

48 

47  37 

53 

50 

41  28 

44  18 

39 

46 

49 

43  12 

57 

42  38 

46  5 

42  3 

47  11 

41  28 

48  16 

54 

45  35 

42  8 

45  2 

43  19 

44  29 

44  29 

54 

45  38 

43  19 

45 

44 

51 

42  7 

56 

55 

46  19 

50 

46 

44  ] 

45  11 

45  11 

44  36 

46  20 

44  0 

47  27 

43  24 

48  33 

46 

49  37 

56 

47  3 

43  34 

46  29 

45 

53 

55 

45  17 

47  4 

40 

48  10 

44  3 

49  16 

43  25 

50  20 

57 

46 

44  19 

47  11 

45  30 

46  35 

46  40 

58 

49 

45  20 

55 

42 

50  1 

44  3 

51  5 

58 

48  29 

45  6 

53 

46  17 

47  16 

47  27 

46  38 

48  35 

59 

49  42 

45  20 

47 

40 

51 

59 

49  11 

55 

48  34 

47  6 

56 

48  16 

47  17 

49  24 

46  38 

50  30 

58 

51  35 

45  17 

52  38 

60 

52 

46  46 

49  14 

57 

48  36 

49  6 

56 

50  14 

47  16 

51  20 

46  35 

52  24 

53 

53  27 

61 

50  33 

47  38 

54 

48  50 

49  15, 

59 

48  34 

51  6 

53 

52  12 

47  11 

53  15 

46  28 

54  18 

62 

51  13 

48  33 

50  33 

49  44 

53 

50  53 

49  11 

52  0 

48  29 

53  5 

46 

54  8 

47  3 

55  10 

63 

53 

49  30 

51  12 

50  41 

50  30 

5149 

48 

56 

49  5 

54  0 

48  21 

55  3 

37 

56  4 

64 

52  31 

50  30 

49 

5140 

51  7 

52  48 

50  24 

53  53 

40 

57 

55 

59 

48  10 

59 

65 

53  9 

5131 

52  26 

52  41 

43 

53  48 

59 

54  53 

50  14 

55  56 

49  28 

56  57 

42 

57  56 

66 

46 

52  35 

53  2 

53  44 

52  18 

54  50 

51  33 

55  55 

47 

56  56 

50  1 

57  57 

49  14 

58  55 

67 

54  22 

53  41 

37 

54  49 

52 

55  55 

52  6 

56  58 

51  19 

57  59 

32 

58  58 

44 

59  55 

68 

57 

54  50 

54  11 

55  57 

53  25 

57  1 

38 

58  3 

50 

59  4 

51  2 

60  1 

50  14 

60  57 

69 

55  31 

56  1 

44 

57  7 

57 

58  10 

53  9 

59  11 

52  21 

60  10 

32 

61  6 

43 

62  1 

70 

56  4 

57  15 

55  16 

58  19 

54  28 

59  21 

39 

60  21 

50 

61  18 

52  1 

62  13 

51  10 

63  7 

71 

36 

58  31 

47 

59  34 

58 

60  35 

54  8 

61  33 

53  18 

62  29 

28 

63  22 

37 

64  14 

72 

57  7 

59  50 

56  17 

60  52 

55  27 

61  51 

36 

62  47 

45 

63  41 

54 

64  33 

52  3 

65  23 

73 

36 

61  12 

46 

62  12 

55 

63  9 

55  3 

64  3 

54  11 

64  56 

53  19 

65  46 

27 

66  34 

74 

58  4 

62  36 

57  13 

63  34 

56  21 

64  29 

29 

65  21 

36 

66  12 

43 

67  0 

50 

67  46 

75 

31 

64  3 

39 

64  58 

46 

65  51 

53 

66  42 

55  0 

67  30 

54  6 

68  16 

53  12 

69  0 

76 

57 

65  32 

58  4 

66  25 

57  10 

67  16 

56  16 

68  4 

22 

68  50 

28 

69  34 

33 

70  16 

77 

59  21 

67  4 

27 

67  54 

33 

68  43 

38 

69  29 

43 

70  12 

48 

70  54 

53 

71  33 

78 

44 

68  39 

49 

69  26 

54 

70  12 

58 

70  55 

56  3 

71  36 

55  7 

72  15 

54  11 

72  52 

79 

60  5 

70  16 

59  9 

71  0 

58  13 

7143 

57  17 

72  23 

21 

73  1 

25 

73  38 

28 

74  12 

80 

24 

7155 

28 

72  36 

31 

73  16 

35 

73  53 

38 

74  28 

41 

75  2 

44 

75  34 

81 

42 

73  36 

45 

74  14 

48 

74  50 

51 

75  24 

53 

75  57 

56 

76  27 

58 

76  57 

82 

58 

75  20 

60  0 

75  54 

59  3 

76  27 

58  5 

76  57 

57  7 

77  27 

56  9 

77  54 

55  11 

78  21 

83 

61  12 

77  6 

14 

77  36 

16 

78  5 

18 

78  32 

19 

78  58 

21 

79  22 

22 

79  46 

84 

25 

78  53 

26 

79  19 

28 

79  44 

29 

80  8 

30 

80  30 

31 

80  51 

32 

81  12 

85 

36 

80  42 

36 

81  4 

38 

8125 

38 

8145 

39 

82  4 

40 

82  21 

41 

82  38 

86 

44 

82  32 

45 

82  50 

46 

83  7 

46 

83  23 

47 

83  38 

47 

83  52 

48 

84  6 

87 

51 

84  23 

51 

84  36 

52 

84  49 

52 

85  1 

53 

85  13 

53 

85  23 

53 

85  34 

88 

56 

86  15 

56 

86  24 

56 

86  32 

56 

86  40 

57 

86  48 

57 

86  55 

57 

87  2 

89 

59 

88  7 

59 

88  12 

59 

88  16 

59 

88  20 

59 

88  24 

59 

88  27 

59 

88  31 

90 

62  0 

90  0 

61  0 

90  0 

60  0 

90  0 

59  0 

90  0 

58  0 

90  0 

57  0 

90  0 

56  0 

90  0 

302 


Table  13.    Kelvin's  Sumner  Line  Table 


b 

o 

a  =  35° 

a  =  36° 

a  =  37° 

a  =  38° 

a  =  39° 

a  =  40° 

a  =  41° 

K 

Q 

K 

Q 

K 

Q 

K 

Q 

K 

Q 

K 

Q 

K 

Q 

o    / 

o    / 

o    / 

t 

o    / 

o   / 

o    / 

o    / 

o    / 

o   / 

o   / 

o    / 

0    / 

o   / 

0 

0  0 

35  0 

0  0 

36  0 

0  0 

37  0 

0  0 

38  0 

0  0 

39  0 

0  0 

40  0 

0  0 

41  0 

1 

49 

0 

49 

0 

48 

0 

47 

0 

47 

0 

46 

0 

45 

0 

2 

1  38 

1 

1  37 

1 

1  36 

1 

1  35 

1 

1  33 

1 

1  32 

1 

131 

1 

3 

2  27 

2 

2  26 

2 

2  24 

2 

2  22 

2 

2  20 

2 

2  18 

2 

2  16 

2 

4 

3  17 

4 

3  14 

4 

3  12 

4 

3  9 

4 

3  7 

4 

3  4 

4 

3  1 

4 

5 

4  6 

6 

4  3 

6 

59 

6 

56 

6 

53 

6 

50 

6 

46 

6 

6 

55 

9 

51 

9 

4  47 

9 

4  44 

9 

4  40 

9 

4  36 

9 

4  31 

9 

7 

5  44 

12 

5  39 

12 

5  35 

12 

5  31 

12 

5  26 

13 

5  21 

13 

5  17 

13 

8 

6  33 

16 

6  28 

16 

6  23 

16 

6  18 

16 

6  13 

17 

6  7 

17 

6  2 

17 

9 

7  22 

20 

7  16 

20 

7  11 

20 

7  5 

21 

59 

21 

53 

21 

47 

21 

10 

8  11 

25 

8  5 

25 

58 

25 

52 

26 

7  45 

26 

7  39 

26 

7  32 

26 

11 

9  0 

30 

53 

30 

8  46 

31 

8  39 

31 

8  32 

31 

8  24 

31 

8  17 

32 

12 

48 

36 

9  41 

36 

9  34 

37 

9  26 

37 

9  18 

37 

9  10 

37 

9  2 

38 

13 

10  37 

42 

10  29 

43 

10  21 

43 

10  13 

43 

10  4 

44 

55 

44 

47 

44 

14 

11  26 

49 

11  17 

50 

11  8 

50 

59 

50 

50 

51 

10  41 

51 

10  31 

51 

15 

12  14 

56 

12  5 

57 

56 

57 

11  46 

58 

11  36 

59 

11  26 

59 

11  16 

59 

16 

13  3 

36  4 

53 

37  5 

12  43 

38  5 

12  33 

39  6 

1^22 

40  7 

12  11 

41  7 

12  0 

42  7 

17 

51 

13 

13  41 

13 

13  30 

14 

13  19 

15 

13  8 

16 

56 

16 

45 

16 

18 

14  40 

22 

14  29 

22 

14  17 

23 

14  6 

24 

54 

25 

13  41 

25 

13  29 

26 

19 

15  28 

31 

15  16 

32 

15  4 

33 

52 

34 

14  40 

35 

14  26 

35 

14  13 

36 

20 

16  16 

41 

16  4 

43 

51 

44 

15  38 

44 

15  25 

45 

15  11 

46 

57 

46 

21 

17  4 

52 

51 

54 

16  38 

55 

16  24 

55 

16  10 

56 

56 

57 

15  41 

57 

22 

52 

37  4 

17  38 

38  5 

17  25 

39  6 

17  10 

40  7 

55 

41  8 

16  41 

42  9 

16  25 

43  9 

23 

18  40 

16 

18  25 

17 

18  11 

18 

56 

19 

17  40 

20 

17  25 

21 

17  9 

22 

24 

19  28 

28 

19  12 

30 

57 

31 

18  42 

32 

18  25 

33 

18  9 

34 

53 

35 

25 

20  15 

41 

59 

43 

19  43 

45 

19  27 

46 

19  10 

47 

53 

48 

18  36 

48 

26 

21  3 

55 

20  46 

57 

20  29 

59 

20  13 

41  0 

55 

42  1 

19  37 

43  2 

19  19 

44  3 

27 

50 

38  10 

21  33 

39  12 

21  15 

40  13 

58 

15 

20  40 

16 

20  21 

17 

20  2 

18 

28 

22  37 

25 

22  19 

27 

22  1 

29 

21  43 

30 

21  24 

32 

21  5 

33 

45 

33 

29 

23  24 

41 

23  5 

43 

47 

45 

22  28 

40 

22  8 

48 

48 

49 

21  28 

49 

30 

24  11 

57 

51 

40  0 

23  32 

41  2 

23  12 

42  3 

52 

43  5 

22  31 

44  6 

22  10 

45  6 

31 

57 

39  15 

24  37 

17 

24  17 

19 

57 

21 

23  36 

22 

23  14 

23 

52 

24 

32 

25  44 

33 

25  23 

35 

25  2 

37 

24  41 

39 

24  19 

41 

57 

42 

23  34 

43 

33 

26  30 

52 

26  9 

54 

47 

56 

25  25 

58 

25  2 

44  0 

24  40 

45  1 

24  16 

46  2 

34 

27  16 

40  11 

54 

41  14 

26  32 

42  16 

26  9 

43  18 

45 

20 

25  22 

21 

58 

22 

35 

28  2 

31 

27  39 

34 

27  16 

.  37 

52 

39 

26  28 

40 

26  4 

41 

25  39 

42 

36 

47 

52 

28  24 

56 

28  0 

58 

27  35 

44  0 

27  11 

45  2 

46 

46  3 

26  20 

47  3 

37 

29  32 

41  14 

29  8 

42  18 

44 

43  20 

28  18 

22 

53 

24 

27  27 

25 

27  1 

25 

38 

30  17 

37 

52 

41 

29  27 

43 

29  1 

45 

28  35 

47 

28  8 

48 

41 

48 

39 

31  2 

42  1 

30  36 

43  4 

30  10 

44  7 

44 

45  9 

29  17 

46  11 

49 

47  12 

28  21 

48  12 

40 

46 

26 

31  20 

29 

53 

32 

30  26 

34 

58 

35 

29  30 

36 

29  1 

37 

41 

32  30 

51 

32  3 

55 

31  36 

58 

31  8 

46  0 

30  39 

47  1 

30  10 

48  2 

41 

49  2 

42 

33  14 

43  18 

46 

44  21 

32  18 

45  24 

49 

26 

31  20 

28 

50 

28 

30  20 

28 

43 

58 

45 

33  29 

49 

33  0 

52 

32  30 

53 

32  0 

55 

31  30 

55 

59 

55 

44 

34  41 

44  14 

34  12 

45  17 

42 

46  20 

33  11 

47  22 

40 

48  23 

32  9 

49  24 

31  37 

50  23 

45 

35  24 

43 

54 

46 

34  23 

49 

52 

51 

33  20 

52 

48 

53 

32  15 

52 

Table  13.    Kelvin's  Sumner  Line  Table 


303 


b 

a  =  35° 

a  =  36° 

a  =  37° 

a  =  38° 

a  =  39° 

a  =  40° 

a  =  41° 

K 

Q 

K 

Q 

K 

Q 

K 

Q 

K 

Q 

K 

Q 

K 

Q 

o 

o   / 

o   / 

o   / 

0    / 

o 

o    / 

o    / 

O     / 

o 

o   / 

o   / 

O          1 

o   / 

o   / 

45 

35  24 

44  43 

34  54 

45  46 

34  23 

46  49 

33  52 

47  51 

33  2C 

48  52 

32  48 

49  53 

32  15 

50  52 

46 

36  6 

45  14 

35  35 

46  17 

35  4 

47  20 

34  32 

48  22 

59 

49  23 

33  26 

50  23 

53 

51  22 

47 

48 

45 

36  16 

49 

44 

51 

35  12 

53 

34  38 

54 

34  4 

54 

33  30 

53 

48 

37  30 

46  18 

57 

47  21 

36  24 

48  24 

51 

49  25 

35  17 

50  26 

42 

51  26 

34  7 

52  25 

49 

38  11 

52 

37  38 

55 

37  4 

57 

36  30 

59 

55 

59 

35  19 

59 

43 

57 

50 

52 

47  27 

38  18 

48  30 

43 

49  32 

37  8 

50  33 

36  32 

51  33 

56 

52  33 

35  19 

53  31 

51 

39  32 

48  3 

57 

49  6 

38  22 

50  8 

46 

51  9 

37  9 

52  9 

36  32 

53  8 

55 

54  6 

52 

40  12 

41 

39  36 

43 

39  0 

45 

38  23 

46 

46 

45 

37  8 

44 

36  30 

42 

53 

52 

49  19 

40  15 

50  22 

38 

51  23 

39  0 

52  24 

38  22 

53  23 

43 

54  21 

37  4 

55  18 

54 

41  31 

59 

53 

51  2 

40  15 

52  3 

36 

53  3 

57 

54  2 

38  18 

59 

38 

56 

55 

42  9 

50  41 

41  30 

43 

52 

43 

40  12 

43 

39  32 

41 

52 

55  39 

38  11 

56  35 

56 

47 

51  23 

42  7 

52  25 

41  28 

53  25 

47 

54  24 

40  7 

55  22 

39  26 

56  19 

44 

57  15 

57 

43  24 

52  7 

43 

53  9 

42  3 

54  8 

41  22 

55  7 

41 

56  4 

59 

57  1 

39  16 

56 

58 

44  0 

53 

43  19 

54 

38 

53 

56 

51 

41  14 

48 

40  31 

44 

48 

58  38 

59 

36 

53  40 

54 

54  40 

43  12 

55  39 

42  29 

56  36 

46 

57  33 

41  3 

58  28 

40  19 

59  21 

60 

45  11 

54  28 

44  29 

55  28 

46 

56  26 

43  2 

57  23 

42  18 

58  19 

34 

59  13 

49 

60  6 

61 

46 

55  18 

45  2 

56  17 

44  1.9 

57  15 

34 

58  11 

49 

59  6 

42  4 

59 

41  18 

51 

62 

46  20 

56  10 

35 

57  8 

51 

58  5 

44  5 

59  0 

43  20 

54 

34 

60  47 

47 

61  38 

63 

53 

57  3 

46  7 

58  0 

45  22 

56 

36 

50 

50 

60  43 

43  3 

61  35 

42  15 

62  25 

64 

47  25 

57 

39 

54 

52 

59  49 

45  6 

60  42 

44  19 

6134 

31 

62  25 

43 

63  14 

65 

56 

58  53 

47  9 

59  49 

46  22 

60  43 

35 

61  35 

47 

62  26 

58 

63  16 

43  9 

64  4 

66 

48  27 

59  51 

39 

60  46 

51 

61  39 

46  3 

62  30 

45  14 

63  20 

44  25 

64  8 

35 

55 

67 

56 

60  50 

48  8 

61  44 

47  19 

62  36 

30 

63  26 

40 

64  15 

51 

65  2 

44  0 

65  48 

68 

49  25 

61  51 

36 

62  44 

46 

63  34 

56 

64  23 

46  6 

65  11 

45  16 

57 

24 

66  41 

69 

53 

62  54 

49  3 

63  45 

48  13 

64  34 

47  22 

65  22 

31 

66  8 

40 

66  53 

48 

67  36 

70 

50  20 

63  58 

29 

64  48 

38 

65  35 

46 

66  22 

55 

67  6 

46  3 

67  50 

45  10 

68  31 

71 

46 

65  4 

54 

65  52 

49  2 

66  38 

48  10 

67  23 

47  17 

68  6 

25 

68  48 

32 

69  28 

72 

51  10 

66  11 

50  18 

66  58 

26 

67  42 

33 

68  25 

39 

69  7 

46 

69  47 

52 

70  26 

73 

34 

67  20 

41 

68  5 

48 

68  48 

54 

69  29 

48  0 

70  9 

47  6 

70  47 

46  12 

71  25 

74 

57 

68  31 

51  3 

69  14 

50  9 

69  55 

49  15 

70  34 

20 

71  12 

26 

7149 

30 

72  25 

75 

52  18 

69  43 

24 

70  24 

29 

71  3 

34 

7140 

39 

72  16 

44 

72  51 

48 

73  25 

76 

38 

70  56 

43 

7135 

48 

72  12 

52 

72  48 

57 

73  22 

48  1 

73  55 

47  5 

74  27 

77 

57 

72  11 

52  1 

72  48 

51  6 

73  23 

50  9 

73  56 

49  13 

74  29 

17 

75  0 

20 

75  30 

78 

53  15 

73  28 

18 

74  2 

22 

74  35 

25 

75  6 

29 

75  30 

32 

76  5 

35 

76  33 

79 

31 

74  45 

34 

75  17 

37 

75  47 

40 

76  17 

43 

76  45 

46 

77  11 

48 

77  37 

80 

46 

76  4 

49 

76  33 

51 

77  1 

54 

77  28 

56 

77  54 

58 

78  18 

48  0 

78  42 

81 

54  0 

77  24 

53  2 

77  51 

52  4 

78  16 

51  6 

78  41 

50  8 

79  4 

49  10 

79  26 

12 

79  48 

82 

13 

78  46 

14 

79  9 

16 

79  32 

17 

79  54 

19 

80  15 

20 

80  35 

22 

80  54 

83 

24 

80  8 

25 

80  29 

26 

80  49 

27 

81  8 

29 

81  27 

29 

8144 

31 

82  1 

84 

33 

8131 

34 

8149 

35 

82  6 

36 

82  23 

37 

82  39 

37 

82  54 

39 

83  9 

85 

41 

82  54 

42 

83  10 

43 

83  24 

43 

83  38 

44 

83  52 

44 

84  4 

45 

84  17 

86 

48 

84  19 

49 

84  31 

49 

84  43 

49 

84  54 

50 

85  5 

50 

85  15 

50 

S5  25 

87 

53 

85  44 

54 

85  53 

54 

86  2 

54 

86  10 

54 

86  18 

54 

86  26 

54 

86  33 

88 

57 

87  9 

57 

87  15 

57 

87  21 

57 

87  26 

57 

87  32 

57 

87  37 

57 

87  42 

89 

59 

88  34 

59 

88  37 

59 

88  40 

59 

88  43 

59 

88  46 

59 

88  48 

59 

88  51 

90 

55  0 

90  0 

54  0 

90  0 

53  0 

90  0 

52  0 

90  0 

51  0 

90  0 

50  0 

90  0 

49  0 

90  0 

i  Hi-i    Tr--,'--i^Ti 


304 


Table  13.    Kelvin's  Sumner  Line  Table 


b 

o 

a  =  43° 

a  =  43° 

a  =  44° 

a  =  45° 

a  =  46° 

a  =  47° 

a  =  48° 

K 

Q 

K 

Q 

K 

Q 

K 

Q 

K 

Q 

K 

Q 

K 

Q 

o   / 

o   / 

o        1 

o   / 

0    / 

o   / 

o   / 

o    / 

o   / 

o    / 

o    / 

o    , 

o   / 

0    I 

0 

0  0 

42  0 

0  0 

43  0 

0  0 

44  0 

0  0 

45  0 

0  0 

46  0 

0  0 

47  0 

0  0 

48  0 

1 

45 

0 

44 

0 

43 

0 

42 

0 

42 

0 

41 

0 

40 

0 

2 

1  29 

1 

1  28 

1 

1  26 

1 

1  25 

1 

1  23 

1 

1  22 

1 

1  20 

1 

3 

2  14 

2 

2  12 

2 

2  10 

2 

2  7 

2 

2  5 

2 

2  3 

2 

2  0 

2 

4 

58 

4 

55 

4 

53 

4 

50 

4 

47 

4 

44 

4 

40 

4 

5 

3  43 

6 

3  39 

6 

3  36 

6 

3  32 

6 

3  28 

6 

3  25 

6 

3  20 

6 

6 

4  27 

9 

4  23 

9 

4  19 

9 

4  14 

9 

4  10 

9 

4  5 

9 

4  0 

9 

7 

5  12 

13 

5  7 

13 

5  2 

13 

57 

13 

51 

13 

46 

13 

40 

13 

8 

56 

17 

51 

17 

45 

17 

5  39 

17 

5  33 

17 

5  27 

17 

5  20 

17 

9 

6  41 

21 

6  34 

21 

6  28 

21 

6  21 

21 

6  14 

21 

6  7 

21 

6  0 

21 

10 

7  25 

26 

7  18 

26 

7  11 

26 

7  3 

26 

56 

26 

48 

26 

40 

26 

11 

8  9 

32 

8  1 

32 

53 

32 

45 

32 

7  37 

32 

7  29 

32 

7  20 

32 

12 

53 

38 

45 

38 

8  36 

38 

8  27 

38 

8  18 

38 

8  9 

38 

8  0 

38 

13 

9  37 

45 

9  28 

45 

9  19 

45 

9  9 

45 

59 

45 

49 

45 

39 

44 

14 

10  21 

52 

10  11 

52 

10  1 

52 

51 

52 

9  40 

52 

9  30 

52 

9  19 

51 

15 

11  5 

59 

55 

44  0 

44 

45  0 

10  33 

46  0 

10  21 

47  0 

10  10 

48  0 

58 

59 

16 

49 

43  8 

11  38 

8 

11  26 

8 

11  15 

8 

11  2 

8 

50 

8 

10  38 

49  7 

17 

12  33 

17 

12  21 

17 

12  8 

17 

56 

17 

43 

17 

1130 

17 

11  17 

16 

18 

13  17 

26 

13  4 

26 

50 

26 

12  37 

26 

12  24 

26 

12  10 

26 

56 

25 

19 

14  0 

36 

46 

36 

13  32 

36 

13  18 

36 

13  4 

36 

50 

36 

12  35 

35 

20 

44 

47 

14  29 

47 

14  14 

47 

59 

47 

45 

47 

13  29 

46 

13  14 

46 

21 

15  27 

58 

15  12 

58 

56 

58 

14  40 

58 

14  25 

58 

14  9 

57 

53 

57 

22 

16  10 

44  10 

54 

45  10 

15  38 

46  10 

15  21 

47  10 

15  5 

48  10 

4S 

49  9 

14  31 

50  9 

23 

53 

22 

16  36 

22 

16  19 

22 

16  2 

22 

45 

22 

15  27 

21 

15  9 

21 

24 

17  36 

35 

17  18 

35 

17  1 

35 

43 

35 

16  25 

35 

16  0 

34 

47 

34 

25 

18  19 

49 

18  0 

49 

42 

49 

17  23 

49 

17  5 

49 

45 

48 

16  25 

47 

26 

19  1 

45  3 

42 

46  3 

18  23 

47  3 

18  3 

48  3 

44 

49  3 

17  24 

50  2 

17  3 

51  1 

27 

43 

18 

19  24 

18 

19  4 

18 

43 

18 

18  23 

17 

18  2 

17 

41 

16 

28 

20  25 

34 

20  5 

34 

44 

34 

19  23 

33 

19  2 

33 

40 

32 

18  19 

31 

29 

21  7 

50 

46 

50 

20  25 

50 

20  3 

49 

41 

49 

19  18 

48 

56 

47 

30 

49 

46  7 

21  27 

47  7 

21  5 

48  7 

42 

49  6 

20  20 

50  6 

56 

51  5 

19  33 

52  3 

31 

22  30 

25 

22  8 

25 

45 

25 

21  21 

24 

58 

23 

20  34 

22 

20  10 

20 

32 

23  11 

43 

48 

43 

22  25 

43 

22  0 

42 

21  36 

41 

21  11 

40 

46 

38 

33 

52 

47  2 

23  28 

48  2 

23  4 

49  2 

39 

50  1 

22  14 

51  0 

48 

58 

21  22 

56 

34 

24  33 

22 

24  8 

22 

43 

21 

23  18 

20 

52 

19 

22  25 

52  17 

58 

53  15 

35 

25  14 

42 

48 

42 

24  22 

42 

56 

41 

23  29 

39 

23  2 

37 

22  34 

35 

36 

54 

48  4 

25  28 

49  3 

25  1 

50  3 

24  34 

51  2 

24  6 

52  0 

38 

58 

23  10 

56 

37 

26  34 

26 

26  7 

25 

39 

25 

25  11 

23 

43 

22 

24  14 

53  19 

45 

54  17 

38 

27  14 

49 

46 

48 

26  17 

47 

48 

46 

25  19 

44 

50 

41 

24  20 

39 

39 

53 

49  12 

27  24 

50  12 

55 

51  11 

26  25 

52  9 

55 

53  7 

25  25 

54  4 

54 

55  1 

40 

28  32 

37 

28  2 

36 

27  32 

35 

27  2 

33 

26  31 

31 

26  0 

28 

25  28 

24 

41 

29  11 

50  2 

40 

51  1 

28  9 

52  0 

38 

58 

27  7 

55 

35 

52 

26  2 

48 

42 

49 

28 

29  18 

27 

46 

25 

28  14 

53  23 

42 

54  20 

27  9 

55  17 

36 

56  13 

43 

30  27 

55 

55 

54 

29  23 

52 

50 

49 

28  17 

46 

43 

42 

27  9 

38 

44 

31  5 

51  23 

30  32 

52  21 

59 

53  19 

29  25 

54  16 

51 

55  13 

28  17 

56  9 

42 

57  4 

45 

42 

51 

31  9 

50 

30  34 

47 

30  0 

44 

29  25 

40 

50 

36 

28  14 

31 

Table  13.    Kelvin's  Sumner  Line  Table 


305 


b 

a  =  42° 

a  =  43° 

a  =  44° 

a  =  45° 

a  =  46° 

a  =  47° 

a  =  48° 

K 

Q 

K 

Q 

K 

Q 

K 

Q 

K 

Q 

K 

Q 

K 

Q 

o 

o   / 

o   / 

o    / 

o   / 

o   / 

o   / 

o   / 

o   , 

o   , 

o   / 

o   / 

o   / 

o   / 

o    / 

45 

31  42 

51  51 

31  9 

52  50 

30  34 

53  47 

30  0 

54  44 

29  25 

55  40 

28  50 

56  36 

28  14 

57  31 

4G 

32  19 

52  21 

45 

53  19 

31  10 

54  16 

34 

55  13 

59 

56  9 

29  23 

57  4 

46 

59 

47 

55 

52 

32  20 

49 

45 

46 

31  8 

42 

30  32 

38 

55 

33 

29  18 

58  27 

48 

33  31 

53  23 

55 

54  20 

32  19 

55  17 

42 

56  13 

31  5 

57  8 

30  27 

58  2 

49 

56 

49 

34  7 

55 

33  30 

52 

53 

49 

32  15 

44 

37 

39 

59 

33 

30  20 

59  26 

50 

42 

54  29 

34  4 

55  25 

33  26 

56  21 

48 

57  16 

32  9 

58  10 

31  30 

59  4 

50 

56 

51 

35  17 

55  3 

38 

59 

59 

55 

33  20 

49 

40 

43 

32  0 

36 

31  20 

60  28 

52 

51 

38 

35  11 

56  34 

34  32 

57  29 

52 

58  23 

33  11 

59  16 

30 

60  8 

49 

61  0 

53 

36  24 

56  15 

44 

57  10 

35  4 

58  4 

34  23 

58 

42 

50 

33  0 

42 

32  18 

33 

54 

57 

52 

36  17 

47 

35 

40 

54 

59  33 

34  12 

60  25 

29 

61  16 

46 

62  7 

55 

37  30 

57  30 

48 

58  24 

36  6 

59  17 

35  24 

60  10 

41 

61  1 

58 

52 

33  14 

41 

56 

38  2 

58  9 

37  19 

59  3 

36 

56 

53 

47 

35  10 

38 

34  26 

62  28 

41 

63  17 

57 

33 

50 

50 

43 

37  6 

60  35 

36  22 

61  26 

38 

62  15 

53 

63  4 

34  8 

53 

58 

39  4 

59  31 

38  20 

60  24 

35 

61  15 

51 

62  5 

36  6 

54 

35  20 

42 

34 

64  30 

59 

34 

60  14 

49 

61  5 

38  4 

56 

37  19 

45 

33 

63  33 

46 

64  21 

35  0 

65  7 

60 

40  4 

57 

39  18 

48 

32 

62  38 

46 

63  26 

59 

64  14 

36  12 

65  0 

25 

46 

01 

33 

61  42 

46 

62  32 

59 

63  21 

38  12 

64  8 

37  25 

55 

37 

40 

49 

66  25 

62 

41  1 

62  28 

40  13 

63  17 

39  26 

64  4 

38 

51 

50 

65  37 

37  1 

66  21 

36  13 

67  5 

63 

28 

63  15 

40 

64  2 

52 

49 

39  3 

65  35 

38  14 

66  20 

25 

67  3 

36 

46 

64 

54 

64  2 

41  6 

49 

40  17 

65  35 

27 

66  20 

38 

67  3 

48 

46 

58 

68  28 

65 

42  20 

51 

31 

65  37 

41 

66  22 

51 

67  5 

39  1 

48 

38  10 

68  30 

37  20 

69  10 

66 

45 

65  41 

55 

66  26 

41  5 

67  10 

40  14 

52 

23 

68  33 

32 

69  14 

41 

53 

67 

43  10 

66  32 

42  19 

67  16 

28 

58 

36 

68  40 

45 

69  20 

53 

59 

38  1 

70  37 

68 

33 

67  25 

42 

68  7 

50 

68  48 

58 

69  28 

40  6 

70  7 

39  13 

70  45 

21 

71  22 

69 

56 

68  18 

43  4 

59 

42  11 

69  38 

41  19 

70  17 

26 

55 

33 

71  31 

40 

72  7 

70 

44  18 

69  12 

25 

69  52 

31 

70  30 

39 

71  7 

45 

7143 

51 

72  19 

58 

53 

71 

39 

70  7 

45 

70  45 

51 

71  22 

58 

58 

41  3 

72  33 

40  9 

73  7 

39  15 

73  40 

72 

59 

71  3 

44  4 

71  40 

43  10 

72  15 

42  16 

72  50 

21 

73  23 

26 

56 

31 

74  27 

73 

45  18 

72  1 

22 

72  36 

28 

73  9 

33 

73  42 

38 

74  14 

42 

74  45 

47 

75  15 

74 

36 

59 

40 

73  32 

45 

74  4 

49 

74  35 

54 

75  6 

58 

75  35 

40  2 

76  4 

75 

53 

73  58 

57 

74  29 

44  1 

75  0 

43  5 

75  29 

42  9 

58 

41  12 

76  26 

16 

53 

76 

46  9 

74  58 

45  12 

75  27 

16 

56 

19 

76  24 

23 

76  51 

26 

77  17 

29 

77  43 

77 

24 

75  58 

27 

76  26 

30 

76  53 

33 

77  19 

36 

77  45 

39 

78  9 

41 

78  33 

78 

38 

77  0 

41 

77  26 

43 

77  51 

46 

78  15 

48 

78  39 

51 

79  2 

53 

79  24 

79 

51 

78  2 

53 

78  26 

55 

78  49 

57 

79  12 

43  0 

79  34 

42  2 

55 

41  3 

80  15 

80 

47  3 

79  5 

46  5 

79  27 

45  6 

79  48 

44  8 

80  9 

10 

80  29 

12 

80  48 

13 

81  7 

81 

13 

80  9 

15 

80  29 

16 

80  48 

18 

81  7 

20 

81  25 

21 

8142 

22 

59 

82 

23 

81  13 

24 

81  31 

26 

81  48 

27 

82  5 

28 

82  21 

29 

82  36 

30 

82  52 

83 

32 

82  18 

32 

82  33 

34 

82  48 

34 

83  3 

35 

83  17 

36 

83  31 

37 

83  45 

84 

39 

83  23 

40 

83  36 

41 

83  49 

.  41 

84  2 

42 

84  14 

42 

84  26 

43 

84  38 

85 

46 

84  28 

46 

84  39 

47 

84  50 

47 

85  1 

47 

85  11 

48 

85  21 

48 

85  31 

86 

51 

85  34 

51 

85  43 

51 

85  52 

52 

86  0 

52 

86  9 

52 

86  17 

52 

86  24 

87 

55 

86  40 

55 

86  47 

55 

86  54 

55 

87  0 

55 

87  6 

56 

87  12 

56 

87  18 

88 

58 

87  47 

58 

87  51 

58 

87  56 

58 

88  0 

58 

88  4 

58 

88  8 

58 

88  12 

89 

59 

88  53 

59 

88  56 

59 

88  58 

59 

89  0 

59 

89  2 

43  0 

89  4 

42  0 

89  6 

90 

48  0 

90  0 

47  0 

90  0 

46  0 

90  0 

45  0 

90  0 

44  0 

90  0 

0 

90  0 

0 

90  0 

306 


Table  13.    Kelvin's  Sumner  Line  Table 


b 

o 

a  =  49° 

a  =  50° 

a  =  51° 

a  =  53° 

a  =  53° 

a  =  54° 

a  =  55° 

K 

Q 

K 

Q 

K 

Q 

K 

Q 

K 

Q 

K 

Q 

K 

Q 

o    / 

o    / 

0     / 

o    t 

0     / 

o    / 

o   / 

o    / 

o    / 

o    / 

o    / 

o    / 

o    / 

o    / 

0 

0  0 

49  0 

0  0 

50  0 

0  0 

51  0 

0  0 

52  0 

0  0 

53  0 

0  0 

54  0 

0  0 

55  0 

1 

39 

0 

39 

0 

38 

0 

37 

0 

36 

0 

35 

0 

34 

0 

2 

1  19 

1 

1  17 

1 

1  16 

1 

1  14 

1 

1  12 

1 

1  11 

1 

1  9 

1 

3 

58 

2 

56 

2 

53 

2 

51 

2 

48 

2 

46 

2 

43 

2 

4 

2  37 

4 

2  34 

4 

2  31 

4 

2  28 

4 

2  24 

4 

2  21 

4 

2  18 

4 

5 

3  17 

6 

3  13 

6 

3  9 

6 

3  5 

6 

3  0 

6 

56 

6 

52 

6 

6 

56 

9 

51 

9 

47 

9 

41 

9 

36 

9 

3  31 

9 

3  26 

9 

7 

4  35 

13 

4  30 

13 

4  24 

13 

4  18 

12 

4  12 

12 

4  6 

12 

4  0 

12 

8 

5  14 

17 

5  8 

17 

5  2 

17 

55 

16 

48 

16 

41 

16 

35 

16 

9 

53 

21 

46 

21 

39 

21 

5  32 

21 

5  24 

20 

5  16 

20 

5  9 

20 

10 

6  32 

26 

6  25 

26 

6  16 

26 

6  8 

26 

6  0 

25 

51 

25 

43 

25 

11 

7  11 

32 

7  3 

31 

54 

31 

45 

31 

36 

30 

6  26 

30 

6  17 

30 

12 

50 

38 

41 

37 

7  31 

37 

7  21 

37 

7  11 

36 

7  1 

36 

51 

36 

13 

8  29 

44 

8  19 

44 

8  8 

44 

58 

43 

47 

43 

36 

42 

7  25 

42 

14 

9  8 

51 

57 

51 

45 

51 

8  34 

50 

8  22 

50 

8  11 

49 

59 

49 

15 

47 

59 

9  35 

59 

9  22 

58 

9  10 

58 

58 

57 

45 

56 

8  32 

56 

16 

10  25 

50  7 

10  12 

51  7 

59 

52  6 

46 

53  6 

9  33 

54  5 

9  19 

55  4 

9  6 

56  3 

17 

11  4 

16 

50 

15 

10  36 

15 

10  22 

14 

10  8 

13 

54 

12 

39 

11 

18 

42 

25 

11  28 

24 

11  13 

24 

58 

23 

43 

22 

10  28 

21 

10  13 

20 

19 

12  20 

35 

12  5 

34 

49 

34 

11  34 

33 

11  18 

32 

11  2 

31 

46 

29 

20 

58 

45 

42 

45 

12  26 

44 

12  9 

43 

53 

42 

36 

41 

11  19 

39 

21 

13  36 

56 

13  19 

56 

13  2 

55 

45 

54 

12  27 

52 

12  10 

51 

52 

49 

22 

14  14 

51  8 

56 

52  7 

38 

53  6 

13  20 

54  5 

13  2 

55  3 

43 

56  2 

12  25 

57  0 

23 

51 

20 

14  33 

19 

14  14 

18 

55 

17 

36 

15 

13  17 

14 

57 

12 

24 

15  29 

33 

15  9 

32 

50 

30 

14  30 

29 

14  10 

27 

50 

26 

13  30 

24 

25 

16  6 

46 

46 

45 

15  26 

43 

15  5 

42 

44 

40 

14  23 

38 

14  2 

36 

26 

43 

52  0 

16  22 

59 

16  1 

57 

40 

55 

15  18 

53 

56 

51 

34 

49 

27 

17  20 

14 

58 

53  13 

36 

54  11 

16  14 

55  9 

51 

56  7 

15  29 

57  5 

15  6 

58  2 

28 

56 

29 

17  34 

28 

17  11 

26 

48 

24 

16  25 

22 

16  1 

19 

37 

16 

29 

18  33 

45 

18  10 

44 

46 

42 

17  22 

39 

58 

37 

33 

34 

16  9 

31 

30 

19  9 

53  2 

45 

54  0 

18  20 

58 

56 

55 

17  31 

52 

17  5 

49 

40 

.  46 

31 

45 

19 

19  20 

17 

55 

55  14 

18  29 

56  11 

18  3 

57  8 

37 

58  5 

17  11 

59  2 

32 

20  21 

36 

55 

34 

19  29 

31 

19  2 

28 

36 

25 

18  9 

22 

42 

18 

33 

56 

54 

20  30 

52 

20  3 

49 

35 

46 

19  8 

42 

40 

39 

18  12 

35 

34 

2131 

54  13 

21  4 

55  11 

37 

56  7 

20  8 

57  4 

40 

58  0 

19  11 

56 

42 

52 

35 

22  6 

33 

38 

30 

21  10 

•26 

41 

23 

20  12 

19 

42 

59  14 

19  12 

60  10 

36 

41 

53 

22  12 

50 

43 

46 

21  13 

42 

43 

38 

20  13 

33 

42 

28 

37 

23  15 

55  14 

46 

56  10 

22  16 

57  7 

45 

58  2 

21  14 

58 

43 

52 

20  12 

47 

38 

49 

35 

23  19 

32 

48 

28 

22  17 

23 

45 

59  18 

21  13 

60  12 

41 

61  7 

39 

24  23 

57 

52 

54 

23  20 

49 

48 

44 

22  15 

39 

43 

33 

21  10 

27 

40 

57 

56  20 

24  24 

57  16 

52 

58  11 

23  19 

59  6 

45 

60  0 

22  12 

54 

38 

48 

41 

25  30 

44 

56 

39 

24  23 

34 

50 

29 

23  15 

22 

41 

61  16 

22  6 

62  9 

42 

26  3 

57  8 

25  28 

58  3 

54 

58 

24  20 

52 

45 

45 

23  10 

38 

34 

31 

43 

35 

33 

26  0 

28 

25  25 

59  22 

50 

60  15 

24  14 

61  8 

38 

62  1 

23  2 

53 

44 

27  7 

59 

31 

53 

55 

47 

25  19 

40 

43 

32 

24  6 

25 

29 

63  16 

45 

38 

58  25 

27  2 

59  19 

26  25 

60  12 

48 

61  5 

25  11 

57 

34 

49 

56 

40 

Table  13.    Kelyin's  Sumner  Line  Table 


307 


b 

0 

a  =  49° 

a  =  50° 

a  =  51° 

a  =  53° 

a  =  53° 

a  =  54° 

a  =  55° 

K 

Q 

K 

Q 

K 

Q 

K 

Q 

K 

Q 

K 

Q 

K 

Q 

o    / 

O     / 

0     / 

0     / 

o    / 

O     / 

o    / 

o    / 

o    / 

o    / 

0     / 

0    / 

o    / 

o    / 

45 

27  38 

58  25 

27  2 

59  19 

26  25 

60  12 

25  48 

61  5 

25  11 

61  57 

24  34 

62  49 

23  56 

63  40 

46 

28  9 

52 

32 

46 

55 

38 

26  17 

31 

39 

62  22 

25  1 

63  13 

24  22 

64  4 

47 

40 

59  20 

28  2 

60  13 

27  24 

61  5 

46 

57 

26  7 

48 

28 

38 

48 

29 

48 

29  11 

49 

32 

41 

53 

33 

27  14 

62  24 

34 

63  15 

54 

64  4 

25  14 

54 

49 

41 

60  18 

29  1 

61  10 

28  21 

62  1 

41 

52 

27  1 

42 

26  20 

31 

39 

65  20 

50 

30  10 

48 

30 

40 

49 

30 

28  8 

63  20 

27 

64  9 

46 

58 

26  4 

46 

51 

39 

61  19 

58 

62  10 

29  17 

63  0 

35 

49 

53 

38 

27  11 

65  26 

28 

66  13 

52 

31  8 

51 

30  26 

41 

44 

30 

29  1 

64  19 

28  19 

65  7 

36 

5-4 

52 

41 

53 

36 

62  23 

53 

63  13 

30  10 

64  1 

27 

49 

44 

36 

28  0 

66  23 

27  16 

67  9 

54 

32  3 

56 

31  20 

45 

36 

33 

52 

65  20 

29  8 

66  6 

24 

52 

39 

38 

55 

30 

63  30 

46 

64  18 

31  2 

65  5 

30  17 

52 

32 

37 

47 

67  22 

28  2 

68  7 

56 

57 

64  5 

32  12 

52 

27 

38 

41 

66  24 

56 

67  9 

29  10 

53 

24 

37 

57 

33  23 

40 

37 

65  26 

51 

66  12 

31  5 

57 

30  19 

41 

32 

68  25 

45 

69  8 

58 

48 

65  16 

33  2 

66  2 

32  15 

47 

28 

67  31 

41 

68  14 

54 

57 

29  6 

39 

59 

34  13 

53 

26 

38 

39 

67  22 

51 

68  5 

31  3 

47 

30  15 

69  29 

27 

70  10 

60 

37 

66  31 

50 

67  14 

33  2 

58 

32  13 

40 

25 

69  21 

36 

70  2 

47 

42 

61 

35  1 

67  9 

34  13 

52 

24 

68  34 

35 

69  15 

46 

56 

56 

36 

30  7 

71  15 

62 

24 

48 

35 

68  30 

45 

69  11 

56 

51 

32  6 

70  31 

31  16 

71  10 

26 

48 

63 

46 

68  28 

56 

69  9 

34  6 

49 

33  16 

70  28 

26 

71  7 

35 

45 

44 

72  22 

64 

36  8 

69  8 

35  17 

48 

27 

70  27 

36 

71  6 

45 

43 

53 

72  20 

31  2 

56 

65 

29 

50 

38 

70  28 

47 

71  6 

55 

44 

33  3 

72  20 

32  11 

56 

19 

73  31 

66 

49 

70  32 

58 

71  9 

35  6 

46 

34  13 

72  22 

21 

58 

28 

73  32 

36 

74  6 

67 

37  9 

71  14 

36  17 

51 

24 

72  26 

31 

73  1 

38 

73  36 

45 

74  9 

52 

42 

68 

28 

58 

35 

72  33 

42 

73  7 

48 

41 

55 

74  14 

33  1 

46 

32  8 

75  18 

69 

46 

72  42 

53 

73  16 

59 

49 

35  5 

74  21 

34  11 

53 

17 

75  24 

23 

55 

70 

38  3 

73  27 

37  10 

59 

36  15 

74  31 

21 

75  2 

26 

75  33 

32 

76  3 

37 

76  32 

71 

20 

74  12 

26 

74  43 

31 

75  14 

36 

44 

41 

76  13 

46 

42 

51 

77  9 

72 

36 

58 

41 

75  28 

46 

57 

50 

76  26 

55 

54 

59 

77  21 

33  4 

47 

73 

51 

75  44 

56 

76  13 

37  0 

76  41 

36  4 

77  8 

35  8 

77  35 

34  12 

78  1 

16 

78  26 

74 

39  6 

76  31 

38  10 

59 

13 

77  25 

17 

51 

21 

78  16 

24 

41 

28 

79  5 

75 

19 

77  19 

23 

77  45 

26 

78  10 

29 

78  34 

33 

58 

35 

79  21 

39 

44 

76 

32 

78  7 

35 

78  32 

38 

55 

41 

79  18 

44 

79  40 

46 

80  2 

49 

80  23 

77 

44 

56 

47 

79  19 

49 

79  41 

52 

80  2 

54 

80  23 

56 

43 

59 

81  3 

78 

55 

79  45 

57 

80  6 

59 

80  27 

37  2 

46 

36  4 

81  6 

35  6 

81  25 

34  8 

43 

79 

40  5 

80  35 

39  7 

54 

38  9 

81  13 

11 

8131 

13 

49 

14 

82  7 

16 

82  23 

80 

15 

8125 

16 

81  42 

18 

82  0 

19 

82  16 

21 

82  33 

22 

49 

24 

83  4 

81 

23 

82  15 

24 

82  31 

26 

47 

27 

83  2 

28 

83  17 

29 

83  31 

31 

45 

82 

31 

83  6 

32 

83  20 

33 

83  34 

34 

48 

35 

84  1 

36 

84  13 

37 

84  26 

83 

38 

57 

38 

84  10 

39 

84  22 

40 

84  34 

41 

45 

41 

56 

42 

85  7 

84 

44 

84  48 

44 

59 

45 

85  10 

45 

85  20 

46 

85  30 

46 

85  39 

47 

49 

85 

49 

85  40 

49 

85  49 

49 

58 

50 

86  6 

50 

86  15 

50 

86  23 

51 

86  30 

86 

53 

86  32 

53 

86  39 

53 

86  46 

53 

53 

54 

87  0 

54 

87  6 

54 

87  12 

87 

56 

87  24 

56 

87  29 

56 

87  34 

56 

87  39 

57 

45 

57 

49 

57 

54 

88 

58 

88  16 

58 

88  19 

58 

88  23 

58 

88  26 

59 

88  30 

59 

88  33 

59 

88  36 

89 

41  0 

89  8 

40  0 

89  10 

39  0 

89  11 

38  0 

89  13 

37  0 

89  15 

36  0 

89  16 

35  0 

89  18 

90 

0 

90  0 

0 

90  0 

0 

90  0 

0 

90  0 

0 

90  0 

0 

90  0 

0 

90  0 

308 


Table  13.    Kelvin's  Sumner  Line  Table 


b 

a  =  56° 

a  =  57° 

a  =  58° 

a  = 

59° 

a  = 

60° 

a  = 

61° 

a  = 

62° 

K 

Q 

K 

Q 

K 

Q 

K 

Q 

K 

Q 

K 

Q 

K 

Q 

o 

o    / 

o    / 

o    / 

o    / 

o    / 

o    / 

o    / 

o    / 

0     / 

o    / 

o    / 

o    / 

o    / 

o    / 

0 

0  0 

56  0 

0  0 

57  0 

0  0 

58  0 

0  0 

59  0 

0  0 

60  0 

0  0 

61  0 

0  0 

62  0 

1 

34 

0 

33 

0 

32 

0 

31 

0 

30 

0 

29 

0 

28 

0 

2 

1  7 

1 

1  5 

1 

1  4 

1 

1  2 

1 

1  0 

1 

58 

1 

56 

1 

3 

41 

2 

38 

2 

35 

2 

33 

2 

30 

2 

1  27 

2 

1  25 

2 

4 

2  14 

4 

2  11 

4 

2  7 

4 

2  4 

4 

2  0 

4 

56 

4 

53 

3 

5 

48 

6 

43 

6 

39 

6 

34 

6 

30 

6 

2  25 

6 

2  21 

5 

6 

3  21 

9 

3  10 

9 

3  11 

8 

3  5 

8 

3  0 

8 

54 

8 

49 

8 

7 

54 

12 

48 

12 

42 

11 

36 

11 

30 

11 

3  23 

11 

3  17 

11 

8 

4  28 

16 

4  21 

15 

4  14 

15 

4  7 

15 

59 

14 

52 

14 

45 

14 

9 

5  1 

20 

53 

19 

45 

19 

37 

19 

4  29 

18 

4  21 

18 

4  13 

18 

10 

34 

24 

5  26 

24 

5  17 

24 

5  8 

23 

59 

23 

50 

22 

41 

22 

11 

6  7 

29 

58 

29 

48 

29 

38 

28 

5  29 

28 

5  19 

27 

5  8 

26 

12 

40 

35 

6  30 

35 

6  20 

34 

6  9 

33 

58 

33 

47 

32 

36 

31 

13 

7  13 

41 

7  2 

41 

51 

40 

39 

39 

6  28 

38 

6  16 

38 

6  4 

37 

14 

46 

48 

34 

47 

7  22 

46 

7  9 

45 

57 

44 

44 

44 

31 

43 

15 

8  19 

55 

8  6 

54 

53 

53 

39 

52 

7  26 

51 

7  13 

50 

59 

49 

16 

52 

57  3 

38 

58  2 

8  24 

59  0 

8  9 

59 

55 

58 

41 

57 

7  26 

56 

17 

9  25 

11 

9  10 

10 

55 

8 

39 

60  7 

8  24 

61  6 

8  9 

62  4 

53 

63  3 

18 

57 

19 

42 

18 

9  26 

17 

9  9 

15 

53 

14 

37 

12 

8  20 

11 

19 

10  29 

28 

10  13 

27 

56 

26 

39 

24 

9  22 

22 

9  5 

20 

47 

19 

20 

11  1 

38 

44 

36 

10  27 

35 

10  9 

33 

51 

31 

33 

29 

9  14 

27 

21 

33 

48 

11  15 

40 

57 

45 

38 

43 

10  19 

40 

10  0 

38 

41 

36 

22 

12  5 

59 

46 

57 

11  27 

55 

11  7 

53 

48 

50 

28 

48 

10  8 

45 

23 

37 

58  10 

12  17 

59  8 

57 

60  0 

36 

61  3 

11  16 

62  1 

55 

58 

34 

55 

24 

13  9 

22 

48 

19 

12  27 

17 

12  5 

14 

44 

12 

11  22 

63  9 

11  0 

64  5 

25 

40 

34 

13  19 

31 

57 

29 

34 

26 

12  12 

23 

49 

20 

26 

16 

26 

14  11 

47 

49 

44 

13  26 

41 

13  3 

38 

40 

35 

12  16 

31 

52 

27 

27 

42 

59  0 

14  19 

57 

55 

54 

31 

50 

13  7 

47 

43 

43 

12  18 

39 

28 

15  13 

14 

49 

60  10 

14  24 

61  7 

59 

62  3 

34 

59 

13  9 

55 

44 

51 

29 

44 

28 

15  19 

24 

53 

21 

14  27 

17 

14  1 

63  12 

36 

64  8 

13  9 

65  4 

30 

16  14 

43 

48 

39 

15  22 

35 

55 

31 

28 

26 

14  2 

21 

34 

17 

31 

44 

58 

16  17 

54 

50 

50 

15  23 

45 

55 

40 

28 

35 

59 

30 

32 

17  14 

60  14 

46 

61  10 

16  18 

62  5 

50 

63  0 

15  22 

55 

53 

49 

14  24 

44 

33 

44 

30 

17  15 

26 

46 

21 

mil 

15 

48 

64  10 

15  19 

65  4 

49 

58 

34 

18  13 

47 

44 

42 

17  14 

37 

44 

31 

16  14 

25 

44 

19 

15  13 

66  13 

35 

42 

61  5 

18  12 

59 

42 

.  54 

17  11 

48 

40 

41 

16  9 

35 

37 

28 

36 

19  11 

23 

40 

62  17 

18  9 

63  11 

37 

64  5 

17  6 

58 

34 

51 

16  1 

44 

37 

40 

41 

19  8 

35 

36 

29 

18  3 

22 

31 

65  15 

58 

66  7 

25 

67  0 

38 

20  8 

62  0 

36 

54 

19  3 

47 

29 

40 

56 

32 

17  22 

24 

48 

10 

39 

36 

20 

20  3 

63  13 

29 

64  6 

55 

58 

18  21 

50 

46 

42 

17  11 

33 

40 

21  4 

40 

30 

33 

55 

25 

19  20 

65  17 

45 

66  8 

18  10 

67  0 

34 

50 

41 

31 

63  1 

56 

53 

20  21 

45 

45 

36 

19  9 

27 

33 

18 

56 

68  S 

42 

58 

23 

21  22 

64  14 

46 

65  5 

20  10 

56 

33 

47 

56 

37 

18  18 

26 

43 

22  25 

45 

48 

36 

21  11 

26 

34 

66  17 

56 

67  7 

19  19 

56 

40 

45 

44 

51 

64  7 

22  14 

58 

36 

48 

58 

38 

20  19 

27 

41 

68  16 

19  2 

69  4 

45 

23  17 

30 

39 

65  20 

22  0 

66  10 

21  22 

59 

42 

48 

20  3 

36 

23 

24 

Table  13.    Kelvin's  Sumner  Line  Table 


309 


b 

a  =  56° 

a  =  57° 

a  =  58° 

a  =  59° 

a  =  60° 

a  =  61° 

a  =  62° 

K 

Q 

K 

Q 

K 

Q 

K 

Q 

K 

Q 

K 

Q 

K 

Q 

c 

o    / 

o    / 

o    / 

o    , 

0     / 

o    / 

o    / 

o    / 

o    / 

0     / 

o    / 

o    / 

o    / 

o   r 

4S 

23  17 

64  30 

22  39 

65  20 

22  0 

66  10 

21  22 

66  59 

20  42 

67  48 

20  3 

68  36 

19  23 

69  24 

46 

43 

54 

23  4 

43 

24 

32 

45 

67  21 

21  5 

68  9 

25 

56 

44 

44 

47 

24  8 

65  18 

28 

66  7 

48 

55 

22  8 

43 

27 

31 

46 

69  17 

20  5 

70  4 

48 

33 

43 

52 

31 

23  11 

67  18 

30 

68  6 

49 

53 

21  7 

39 

25 

25 

49 

58 

66  8 

24  16 

55 

34 

42 

52 

29 

22  10 

69  15 

28 

70  1 

45 

46 

50 

25  22 

34 

39 

67  20 

57 

68  7 

23  14 

53 

31 

38 

48 

23 

21  5 

71  8 

51 

46 

67  0 

25  2 

46 

24  19 

32 

36 

69  17 

52 

70  2 

22  8 

46 

24 

30 

52 

26  9 

27 

25 

68  12 

41 

57 

57 

42 

23  12 

26 

28 

71  9 

43 

52 

53 

32 

54 

47 

39 

25  2 

69  23 

24  17 

70  7 

32 

50 

47 

33 

22  1 

72  15 

54 

54 

68  22 

26  9 

69  6 

23 

50 

37 

33 

52 

71  15 

23  6 

57 

19 

39 

55 

27  16 

51 

30 

34 

44 

70  17 

57 

59 

24  11 

41 

24 

72  22 

37 

73  2 

56 

37 

69  20 

51 

70  2 

26  4 

44 

25  17 

71  26 

29 

72  7 

42 

47 

54 

26 

57 

58 

50 

27  11 

31 

23 

71  12 

36 

53 

47 

33 

24  0 

73  12 

23  11 

51 

58 

28  18 

70  20 

31 

71  1 

42 

41 

54 

72  20 

25  5 

59 

17 

38 

28 

74  16 

59 

38 

51 

50 

31 

27  1 

72  10 

26  12 

48 

23 

73  26 

33 

74  4 

44 

41 

60 

58 

71  22 

28  9 

72  1 

19 

39 

29 

73  17 

40 

54 

49 

30 

59 

75  7 

61 

29  17 

54 

27 

32 

37 

73  9 

46 

46 

56 

74  22 

25  5 

57 

24  14 

33 

62 

35 

72  26 

45 

73  3 

54 

39 

27  3 

74  15 

26  12 

50 

21 

75  25 

29 

59 

63 

53 

59 

29  2 

35 

28  11 

74  10 

19 

45 

27 

75  19 

36 

53 

44 

76  26 

64 

30  10 

73  32 

19 

74  7 

27 

41 

35 

75  15 

42 

48 

50 

76  21 

58 

53 

65 

27 

74  5 

35 

39 

42 

75  12 

50 

45 

57 

76  17 

26  4 

49 

25  11 

77  20 

66 

43 

39 

50 

75  12 

57 

44 

28  4 

76  16 

27  11 

47 

17 

77  18 

24 

48 

67 

59 

75  14 

30  5 

46 

29  12 

76  17 

18 

47 

24 

77  17 

30 

47 

36 

78  16 

68 

31  14 

49 

20 

76  20 

26 

50 

31 

77  19 

37 

48 

43 

78  16 

48 

44 

69 

28 

76  25 

34 

54 

39 

77  23 

44 

51 

50 

78  19 

55 

46 

26  0 

79  13 

70 

42 

77  1 

47 

77  29 

52 

56 

57 

78  23 

28  2 

50 

27  6 

79  16 

11 

42 

71 

55 

37 

31  0 

78  4 

30  4 

78  30 

29  9 

56 

13 

79  21 

17 

46 

21 

80  11 

72 

32  8 

78  14 

12 

39 

16 

79  4 

20 

79  29 

24 

53 

27 

80  17 

31 

40 

73 

20 

51 

23 

79  15 

27 

39 

30 

80  2 

34 

80  25 

37 

48 

41 

81  10 

74 

31 

79  28 

34 

51 

37 

80  14 

40 

36 

44 

57 

46 

81  19 

50 

40 

75 

42 

80  6 

44 

80  27 

47 

49 

50 

81  10 

53 

81  30 

55 

50 

58 

82  10 

76 

52 

44 

54 

81  4 

56 

81  24 

59 

44 

29  1 

82  3 

28  3 

82  22 

27  6 

40 

77 

33  1 

81  22 

32  3 

41 

31  5 

82  0 

30  7 

82  18 

9 

36 

11 

54 

13 

83  11 

78 

10 

82  1 

11 

82  19 

13 

36 

15 

53 

17 

83  9 

18 

83  26 

20 

41 

79 

18 

40 

19 

56 

21 

83  12 

22 

83  28 

24 

43 

25 

58 

26 

84  12 

80 

25 

83  19 

26 

83  34 

28 

48 

29 

84  3 

30 

84  16 

31 

84  30 

32 

43 

81 

32 

59 

32 

84  12 

34 

84  25 

35 

38 

36 

50 

37 

85  3 

37 

85  15 

82 

38 

84  38 

38 

50 

39 

85  2 

40 

85  13 

41 

85  24 

42 

35 

42 

46 

83 

43 

85  18 

43 

85  28 

44 

39 

45 

49 

45 

58 

46 

86  8 

46 

86  18 

84 

47 

58 

48 

86  7 

48 

86  16 

49 

86  24 

49 

86  33 

50 

41 

50 

49 

85 

51 

86  38 

52 

46 

52 

53 

52 

87  0 

52 

87  7 

53 

87  14 

53 

87  21 

86 

54 

87  18 

55 

87  24 

55 

87  30 

55 

36 

55 

42 

55 

47 

56 

52 

87 

57 

59 

57 

88  3 

57 

88  8 

57 

88  12 

57 

88  16 

57 

88  20 

58 

88  24 

88 

59 

88  39 

59 

42 

59 

45 

59 

48 

59 

51 

59 

53 

59 

56 

89 

34  0 

89  19 

33  0 

89  21 

32  0 

89  23 

31  0 

89  24 

30  0 

89  25 

29  0 

89  27 

28  0 

89  28 

90 

0 

90  0 

0 

90  0 

0 

90  0 

0 

90  0 

0 

90  0 

0 

90  0 

0 

90  0 

310 


Table  13.    Kelvin's  Sumner  Line  Table 


b 

a  = 
K 

63° 

Q 

a  = 

64° 

a  = 

65° 

a  = 

66° 

a  = 

67° 

a  = 

68° 

a  = 

69° 

K 

Q 

K 

Q 

K 

Q 

K 

Q 

K 

Q 

K 

Q 

o 

o    / 

o    / 

o    / 

o    / 

0     / 

o    / 

o    / 

o    / 

o    / 

0     / 

o    / 

0    / 

o    / 

o    / 

0 

0  0 

63  0 

0  0 

64  0 

0  0 

65  0 

0  0 

66  0 

0  0 

67  0 

0  0 

68  0 

0  0 

69  0 

1 

27 

0 

26 

0 

25 

0 

24 

0 

23 

0 

22 

0 

22 

0 

2 

54 

1 

53 

1 

51 

1 

49 

1 

47 

1 

45 

1 

43 

1 

3 

1  22 

2 

1  19 

2 

1  16 

2 

1  13 

2 

1  10 

2 

1  7 

2 

1  5 

2 

4 

49 

3 

45 

3 

41 

3 

38 

3 

34 

3 

30 

3 

26 

3 

5 

2  16 

5 

2  11 

5 

2  7 

5 

2  2 

5 

57 

5 

52 

5 

47 

4 

6 

43 

7 

38 

7 

32 

7 

26 

7 

2  20 

7 

2  15 

7 

2  9 

6 

7 

3  10 

10 

3  4 

10 

57 

10 

50 

9 

44 

9 

37 

9 

30 

8 

8 

37 

13 

30 

13 

3  22 

13 

3  15 

12 

3  7 

12 

59 

12 

52 

11 

9 

4  4 

17 

56 

17 

47 

16 

39 

16 

30 

15 

3  22 

15 

3  13 

14 

10 

31 

21 

4  22 

21 

4  12 

20 

4  3 

20 

53 

19 

44 

18 

34 

17 

11 

58 

26 

48 

25 

37 

24 

27 

24 

4  16 

23 

4  6 

22 

55 

21 

12 

5  25 

31 

5  14 

30 

5  2 

29 

51 

28 

39 

27 

28 

26 

4  16 

25 

13 

52 

36 

40 

35 

27 

34 

5  15 

33 

5  2 

32 

50 

31 

37 

29 

14 

6  18 

42 

6  5 

40 

52 

39 

39 

38 

25 

37 

5  12 

36 

58 

34 

15 

45 

48 

31 

46 

6  17 

45 

6  3 

44 

48 

42 

34 

41 

5  19 

39 

16 

7  11 

54 

56 

53 

41 

51 

26 

50 

6  11 

48 

56 

47 

40 

45 

17 

38 

64  1 

7  22 

65  0 

7  6 

58 

50 

56 

34 

54 

6  17 

53 

6  1 

51 

18 

8  4 

9 

47 

7 

30 

66  5 

7  13 

67  3 

56 

68  1 

39 

59 

22 

57 

19 

30 

17 

8  12 

15 

54 

12 

37 

10 

7  19 

8 

7  0 

69  6 

42 

70  3 

20 

56 

25 

37 

23 

8  18 

20 

8  0 

18 

41 

15 

22 

13 

7  3 

10 

21 

9  22 

34 

9  2 

31 

42 

28 

23 

26 

8  3 

23 

43 

20 

23 

17 

22 

48 

43 

27 

40 

9  6 

37 

46 

34 

25 

31 

8  4 

28 

43 

24 

23 

10  13 

52 

52 

49 

30 

46 

9  9 

43 

47 

39 

25 

36 

8  3 

32 

24 

39 

65  2 

10  16 

59 

54 

56 

31 

52 

9  9 

48 

46 

44 

23 

40 

25 

11  4 

13 

41 

66  9 

10  17 

67  6 

54 

68  2 

30 

57 

9  7 

53 

43 

49 

26 

29 

24 

11  5 

20 

41 

16 

10  16 

12 

52 

69  7 

27 

70  2 

9  2 

58 

27 

54 

35 

29 

31 

11  4 

26 

38 

22 

10  13 

17 

48 

12 

22 

71  7 

28 

12  19 

47 

53 

42 

27 

37 

11  0 

32 

34 

27 

10  8 

22 

41 

17 

29 

43 

59 

12  16 

54 

50 

49 

22 

43 

55 

38 

28 

32 

10  0 

27 

30 

13  7 

66  11 

40 

67  6 

12  12 

68  1 

44 

55 

11  16 

49 

48 

43 

19 

37 

31 

31 

24 

13  3 

19 

34 

13 

12  6 

69  7 

37 

70  0 

11  8 

54 

38 

47 

32 

55 

38 

26 

32 

56 

25 

27 

19 

57 

12 

27 

71  5 

57 

58 

33 

14  19 

52 

49 

45 

13  18 

38 

48 

31 

12  17 

24 

46 

17 

11  15 

72  9 

34 

43 

67  6 

14  12 

59 

40 

52 

13  9 

44 

37 

37 

12  5 

29 

33 

21 

35 

15  6 

21 

34 

68  13 

14  2 

69-  6 

30 

58 

57 

50 

24 

41 

51 

33 

36 

29 

36 

56 

28 

23 

20 

50 

70  12 

13  17 

71  3 

43 

54 

12  9 

45 

37 

52 

51 

15  18 

43 

44 

34 

14  10 

26 

36 

16 

13  2 

72  7 

27 

57 

38 

16  14 

68  7 

40 

59 

15  5 

49 

30 

40 

55 

30 

20 

20 

45 

73  10 

39 

36 

24 

16  1 

69  15 

26 

70  5 

50 

55 

14  14 

44 

38 

34 

13  2 

23 

40 

58 

41 

22 

31 

46 

21 

15  9 

71  10 

33 

59 

56 

48 

19 

37 

41 

17  20 

58 

43 

48 

16  6 

37 

28 

26 

51 

72  14 

14  14 

73  2 

36 

51 

42 

41 

69  16 

17  4 

70  5 

26 

53 

47 

42 

15  9 

29 

31 

17 

53 

74  5 

43 

18  2 

34 

24 

22 

45 

71  10 

16  6 

58 

27 

45 

48 

32 

14  9 

19 

44 

23 

52 

44 

40 

17  4 

27 

25 

72  14 

45 

73  1 

15  5 

48 

25 

34 

45 

44 

70  11 

18  4 

58 

23 

45 

43 

31 

16  2 

18 

22 

74  3 

41 

49 

Table  13.    Kelvin's  Sumner  Line  Table 


311 


b 

o 

a  =  63° 

a  =  64° 

a  =  65° 

a  =  66° 

a  =  67° 

a  =  68° 

a  =  69° 

K 

Q 

K 

Q 

K 

Q 

K 

Q 

K 

Q 

K 

Q 

K 

Q 

o    / 

o    / 

o    / 

o    / 

o    / 

o    / 

0    / 

o    / 

o    / 

o    / 

o    1 

o   / 

o    / 

o    / 

45 

1844 

70  11 

18  4 

70  58 

17  23 

71  45 

16  43 

72  31 

16  2 

73  18 

15  22 

74  3 

14  41 

74  49 

46 

19  4 

30 

23 

71  17 

42 

72  3 

17  1 

49 

19 

34 

38 

19 

56 

75  4 

47 

24 

50 

42 

36 

18  0 

21 

18 

73  7 

36 

51 

54 

36 

15  11 

20 

48 

43 

71  10 

19  1 

55 

18 

40 

35 

25 

53 

74  8 

16  10 

52 

26 

36 

49 

20  2 

31 

19 

72  15 

36 

59 

52 

43 

17  9 

26 

26 

75  9 

41 

52 

50 

21 

52 

37 

36 

53 

73  19 

18  9 

74  2 

25 

44 

41 

26 

56 

76  8 

51 

40 

72  13 

55 

56 

19  10 

39 

25 

21 

41 

75  3 

56 

44 

16  10 

25 

52 

58 

35 

20  12 

73  17 

27 

59 

41 

40 

56 

21 

17  10 

76  2 

24 

42 

53 

21  16 

57 

29 

38 

43 

74  19 

57 

75  0 

18  11 

40 

24 

20 

38 

59 

54 

33 

73  20 

46 

74  0 

59 

40 

19  13 

20 

26 

59 

38 

38 

51 

77  17 

55 

50 

43 

21  3 

22 

20  15 

75  1 

28 

40 

40 

76  19 

52 

57 

17  4 

35 

56 

22  7 

74  6 

19 

45 

30 

23 

43 

76  1 

54 

39 

18  6 

77  16 

17 

53 

57 

23 

30 

34 

75  7 

45 

45 

57 

22 

19  8 

59 

19 

35 

29 

78  11 

58 

39 

54 

49 

30 

21  0 

76  7 

20  11 

43 

21 

77  19 

32 

55 

41 

30 

59 

54 

75  18 

22  4 

54 

14 

30 

24 

77  5 

34 

40 

44 

78  15 

53 

49 

60 

23  9 

42 

19 

76  18 

28 

53 

37 

27 

47 

78  1 

56 

35 

18  5 

79  8 

61 

24 

76  7 

33 

42 

42 

77  16 

50 

49 

59 

22 

19  8 

55 

16 

27 

62 

38 

33 

46 

77  6 

55 

39 

21  3 

78  11 

20  11 

44 

19 

79  16 

27 

47 

63 

52 

59 

59 

31 

22  7 

78  3 

15 

34 

23 

79  6 

30 

37 

37 

80  7 

64 

24  5 

77  25 

23  12 

56 

19 

27 

27 

57 

34 

28 

41 

58 

47 

27 

65 

18 

51 

24 

78  21 

31 

51 

38 

79  20 

45 

50 

51 

80  19 

57 

47 

66 

30 

78  18 

36 

47 

43 

79  16 

49 

44 

55 

80  12 

20  1 

40 

19  7 

81  8 

67 

42 

45 

48 

79  13 

54 

41 

59 

80  8 

21  5 

35 

10 

81  2 

16 

28 

68 

54 

79  12 

59 

39 

23  4 

80  6 

22  9 

32 

15 

58 

19 

24 

25 

49 

69 

25  5 

39 

24  10 

80  5 

14 

31 

19 

56 

24 

81  21 

28 

46 

33 

82  10 

70 

15 

80  7 

20 

32 

24 

56 

28 

81  20 

33 

44 

37 

82  8 

41 

31 

71 

25 

35 

29 

59 

33 

81  22 

37 

45 

41 

82  8 

45 

30 

49 

53 

72 

35 

81  3 

38 

81  26 

42 

48 

45 

82  10 

49 

32 

52 

53 

56 

83  14 

73 

44 

32 

47 

53 

50 

82  14 

53 

35 

57 

56 

59 

83  16 

20  3 

36 

74 

53 

82  0 

55 

82  20 

58 

40 

23  1 

83  0 

22  4 

83  20 

21  6 

39 

9 

58 

75 

26  1 

29 

25  3 

48 

24  6 

83  7 

8 

26 

11 

44 

13 

84  2 

15 

84  20 

76 

8 

58 

10 

83  16 

13 

34 

15 

51 

17 

84  8 

19 

25 

21 

42 

77 

15 

83  28 

17 

44 

19 

84  1 

21 

84  17 

23 

33 

25 

48 

26 

85  4 

78 

22 

57 

23 

84  13 

25 

28 

27 

43 

28 

57 

30 

85  12 

31 

26 

79 

28 

84  27 

29 

41 

31 

55 

32 

85  9 

33 

85  22 

35 

36 

36 

49 

80 

33 

57 

34 

85  9 

36 

85  22 

37 

35 

38 

47 

39 

59 

40 

86  11 

81 

38 

85  27 

39 

38 

40 

50 

41 

86  1 

42 

86  12 

43 

86  23 

44 

34 

82 

43 

57 

43 

86  7 

44 

86  17 

45 

27 

46 

37 

47 

47 

47 

57 

83 

47 

86  27 

47 

36 

48 

45 

49 

54 

49 

87  2 

50 

87  11 

50 

87  19 

84 

50 

57 

51 

87  5 

51 

87  12 

52 

87  20 

52 

28 

53 

35 

53 

42 

85 

53 

87  27 

54 

34 

54 

40 

54 

47 

54 

53 

55 

59 

55 

88  5 

86 

56 

58 

56 

88  3 

56 

88  8 

56 

88  13 

56 

88  18 

57 

88  23 

57 

28 

87 

58 

88  28 

58 

32 

58 

36 

58 

40 

58 

44 

58 

47 

58 

51 

88 

59 

59 

59 

89  1 

59 

89  4 

59 

89  7 

59 

89  9 

59 

89  12 

59 

89  14 

89 

27  0 

89  29 

26  0 

31 

25  0 

32 

24  0 

33 

23  0 

34 

22  0 

36 

21  0 

37 

90 

0 

90  0 

0 

90  0 

0 

90  0 

0 

90  0 

0 

90  0 

0 

90  0 

0 

90  0 

312 


Table  13.    Kelyin's  Sumner  Line  Table 


b 

a  = 

70° 

a  = 

71° 

a  = 

73° 

a  = 

73° 

a  = 

74° 

a  = 

75° 

a  = 

76° 

K 

Q 

K 

Q 

K 

Q 

K 

Q 

K 

Q 

K 

Q 

K 

Q 

o    / 

o    / 

o   / 

o   / 

0    / 

O    / 

o    / 

o    / 

o    / 

o    / 

o    / 

o    / 

o    / 

o    / 

0 

0  0 

70  0 

0  0 

71  0 

0  0 

72  0 

0  0 

73  0 

0  0 

74  0 

0  0 

75  0 

0  0 

76  0 

1 

21 

0 

20 

0 

19 

0 

18 

0 

17 

0 

16 

0 

15 

0 

2 

41 

1 

39 

1 

37 

1 

35 

1 

33 

1 

31 

1 

29 

0 

3 

1  2 

2 

59 

2 

56 

1 

53 

1 

50 

1 

47 

1 

44 

1 

4 

22 

3 

1  18 

3 

1  14 

2 

1  10 

2 

1  6 

2 

1  2 

2 

58 

2 

5 

43 

4 

38 

4 

33 

4 

28 

4 

23 

3 

18 

3 

1  13 

3 

6 

2  3 

6 

57 

6 

51 

6 

45 

5 

39 

5 

33 

5 

27 

4 

7 

23 

8 

2  17 

8 

2  9 

8 

2  3 

7 

56 

7 

49 

6 

41 

6 

8 

44 

11 

36 

10 

28 

10 

20 

9 

2  12 

9 

2  4 

8 

56 

8 

9 

3  4 

14 

55 

13 

46 

13 

37 

12 

28 

11 

19 

10 

2  10 

10 

10 

24 

17 

3  15 

16 

3  5 

16 

55 

15 

45 

14 

35 

13 

25 

12 

11 

45 

20 

34 

19 

23 

19 

3  12 

18 

3  1 

17 

50 

16 

39 

15 

12 

4  5 

24 

53 

23 

41 

22 

29 

21 

17 

20 

3  5 

19 

53 

18 

13 

25 

28 

4  12 

27 

59 

26 

46 

25 

33 

23 

20 

22 

3  7 

21 

14 

45 

33 

31 

31 

4  17 

30 

4  3 

29 

49 

27 

35 

25 

21 

24 

15 

5  5 

38 

50 

36 

35 

34 

20 

33 

4  5 

31 

50 

29 

35 

27 

16 

25 

43 

5  9 

41 

53 

39 

37 

37 

21 

35 

4  5 

33 

49 

31 

17 

44 

49 

28 

46 

5  11 

44 

54 

42 

37 

40 

20 

38 

4  3 

35 

18 

6  4 

55 

46 

52 

29 

49 

5  11 

47 

53 

45 

35 

42 

17 

40 

19 

24 

71  1 

6  5 

58 

46 

55 

28 

52 

5  9 

50 

50 

47 

31 

44 

20 

43 

7 

24 

72  4 

6  4 

73  1 

44 

58 

25 

55 

5  5 

52 

45 

49 

21 

7  3 

14 

42 

11 

21 

7 

6  1 

74  4 

40 

75  1 

19 

57 

59 

54 

22 

22 

21 

7  0 

18 

39 

14 

17 

10 

56 

7 

34 

76  3 

5  12 

59 

23 

41 

29 

18 

25 

56 

21 

34 

17 

6  11 

13 

48 

9 

26 

77  4 

24 

8  0 

37 

36 

32 

7  13 

28 

50 

24 

26 

19 

6  3 

15 

39 

10 

25 

19 

45 

54 

40 

30 

35 

7  6 

31 

41 

26 

17 

21 

52 

16 

26 

38 

53 

8  12 

48 

47 

43 

22 

38 

56 

33 

31 

27 

6  5 

22 

27 

56 

72  2 

30 

56 

8  4 

51 

38 

46 

7  11 

40 

45 

34 

18 

28 

28 

9  15 

11 

48 

73  5 

21 

59 

54 

54 

26 

48 

59 

41 

31 

35 

29 

33 

20 

9  5 

14 

37 

74  8 

8  9 

75  2 

41 

55 

7  13 

48 

44 

42 

30 

51 

30 

22 

24 

53 

17 

25 

10 

55 

76  3 

26 

56 

57 

49 

31 

10  9 

40 

39 

34 

9  9 

26 

40 

19 

8  10 

11 

40 

77  4 

7  10 

56 

32 

27 

51 

56 

44 

25 

36 

55 

28 

24 

20 

53 

12 

22 

78  4 

33 

44 

73  2 

10  13 

54 

41 

46 

9  10 

37 

38 

29 

8  6 

20 

34 

11 

34 

11  2 

13 

30 

74  4 

57 

56 

25 

46 

52 

38 

19 

28 

46 

19 

35 

19 

24 

46 

15 

10  13 

75  6 

39 

56 

9  6 

47 

32 

37 

58 

27 

36 

36 

36 

11  2 

26 

28 

16 

54 

76  6 

19 

56 

45 

46 

8  10 

36 

37 

53 

48 

18 

37 

43 

27 

10  8 

17 

33 

77  6 

58 

55 

22 

44 

38 

12  9 

74  0 

34 

49 

58 

38 

22 

27 

46 

16 

9  10 

78  4 

34 

53 

39 

26 

12 

50 

75  1 

11  13 

50 

36 

38 

59 

26 

22 

14 

46 

79  2 

40 

42 

25 

12  5 

13 

28 

76  1 

50 

49 

10  12 

37 

34 

24 

57 

11 

41 

58 

38 

20 

26 

42 

13 

11  4 

77  0 

25 

47 

46 

34 

9  8 

21 

42 

13  14 

52 

35 

39 

56 

26 

17 

12 

38 

58 

58 

44 

19 

30 

43 

29 

75  6 

50 

52 

12  10 

38 

30 

24 

50 

78  9 

10  10 

55 

30 

40 

44 

44 

20 

13  4 

76  5 

24 

51 

43 

36 

11  2 

21 

22 

79  5 

41 

50 

45 

59 

34 

18 

19 

38 

77  4 

56 

48 

14 

32 

33 

16 

51 

80  0 

Table  13.    Kelvin's  Sumner  Line  Table 


313 


b 

a  =  70° 

a  =  71° 

a  =  72° 

a  =  73° 

a  =  74° 

a  =  75° 

a  =  76° 

K 

Q 

K 

Q 

K 

Q 

K 

Q 

K 

Q 

K 

Q 

K 

Q 

o 

o    / 

o    / 

O     / 

o    / 

o    / 

o    / 

o    / 

o    / 

o    / 

o    / 

o    / 

o    / 

o    / 

o    / 

45 

13  59 

75  34 

13  18 

76  19 

12  38 

77  4 

1156 

77  48 

11  14 

78  32 

10  33 

79  16 

9  51 

80  0 

46 

14  14 

49 

32 

33 

51 

17 

12  9 

78  1 

26 

44 

44 

27 

10  1 

10 

47 

29 

76  4 

46 

47 

13  4 

30 

21 

13 

38 

56 

55 

39 

11 

21 

48 

43 

19 

14  0 

77  1 

17 

44 

33 

26 

49 

79  8 

11  6 

50 

21 

32 

49 

57 

34 

13 

16 

29 

58 

45 

39 

12  0 

21 

16 

80  2 

31 

43 

50 

15  11 

50 

26 

31 

41 

78  12 

57 

53 

11 

33 

26 

14 

41 

54 

51 

25 

77  6 

39 

46 

53 

26 

13  8 

79  7 

22 

46 

36 

26 

50 

81  5 

52 

38 

22 

52 

78  2 

14  5 

41 

19 

21 

33 

59 

46 

38 

59 

16 

53 

51 

39 

15  4 

18 

17 

56 

30 

35 

43 

80  13 

56 

50 

11  8 

28 

54 

16  4 

55 

16 

34 

29 

79  11 

41 

49 

53 

26 

12  5 

81  3 

17 

40 

55 

16 

78  12 

28 

50 

40 

26 

52 

80  3 

13  3 

40 

14 

16 

26 

52 

56 

28 

30 

40 

79  6 

51 

42 

14  2 

18 

13 

54 

23 

29 

34 

82  4 

57 

40 

47 

51 

23 

15  1 

58 

12 

33 

22 

81  8 

32 

42 

42 

10 

58 

52 

79  5 

16  2 

40 

11 

80  14 

22 

48 

31 

22 

41 

55 

50 

28 

59 

17  3 

23 

12 

57 

21 

30 

31 

81  3 

40 

36 

49 

82  9 

58 

41 

60 

14 

41 

22 

80  14 

31 

46 

40 

18 

49 

50 

57 

22 

12  6 

54 

61 

24 

80  0 

32 

31 

41 

81  3 

49 

34 

57 

82  5 

13  5 

36 

13 

83  7 

62 

34 

18 

42 

49 

50 

20 

58 

50 

14  5 

20 

13 

50 

20 

20 

63 

44 

37 

52 

81  7 

59 

37 

15  6 

82  6 

13 

35 

20 

83  4 

27 

33 

64 

54 

56 

17  1 

25 

16  8 

54 

14 

22 

21 

50 

27 

18 

34 

46 

65 

18  3 

81  15 

10 

43 

16 

82  11 

22 

38 

28 

83  5 

34 

32 

40 

59 

66 

12 

35 

18 

82  2 

24 

28 

30 

55 

35 

21 

41 

47 

46 

84  12 

67 

21 

54 

26 

20 

32 

46 

37 

83  11 

42 

36 

47 

84  1 

52 

26 

68 

29 

82  14 

34 

39 

39 

83  4 

44 

28 

49 

52 

53 

16 

58 

40 

69 

37 

34 

42 

58 

46 

22 

51 

45 

55 

84  8 

59 

31 

13  3 

54 

70 

45 

54 

49 

83  17 

53 

40 

57 

84  2 

15  1 

24 

14  5 

46 

8 

85  8 

71 

52 

83  15 

56 

36 

59 

58 

16  3 

19 

7 

40 

10 

85  1 

13 

22 

72 

59 

35 

18  2 

56 

17  5 

84  16 

9 

36 

12 

56 

15 

16 

18 

36 

73 

19  6 

56 

8 

84  15 

11 

34 

14 

53 

17 

85  12 

20 

31 

23 

50 

74 

12 

84  16 

14 

35 

17 

53 

19 

85  11 

22 

29 

25 

47 

27 

86  4 

75 

18 

37 

20 

54 

22 

85  12 

24 

28 

27 

45 

29 

86  2 

31 

18 

76 

23 

58 

25 

85  14 

27 

30 

29 

46 

31 

86  2 

33 

17 

35 

33 

77 

28 

85  19 

30 

34 

31 

49 

33 

86  4 

35 

19 

37 

33 

38 

47 

78 

33 

40 

34 

54 

35 

86  8 

37 

22 

39 

35 

40 

49 

41 

87  2 

79 

37 

86  2 

38 

86  14 

39 

27 

41 

40 

42 

52 

43 

87  4 

44 

17 

80 

41 

23 

42 

35 

43 

46 

44 

58 

45 

87  9 

46 

20 

47 

31 

81 

45 

44 

45 

55 

46 

87  5 

47 

87  16 

48 

26 

49 

36 

50 

46 

82 

48 

87  6 

48 

87  15 

49 

25 

50 

34 

51 

43 

51 

52 

52 

88  1 

83 

51 

28 

51 

36 

52 

44 

52 

52 

53 

88  0 

53 

88  8 

54 

15 

84 

53 

49 

53 

56 

54 

88  3 

54 

88  10 

55 

17 

55 

24 

56 

30 

85 

55 

88  11 

55 

88  17 

56 

23 

56 

28 

56 

34 

57 

40 

57 

45 

86 

57 

33 

57 

37 

57 

42 

57 

47 

57 

51 

'58 

56 

58 

89  0 

87 

58 

55 

58 

58 

58 

89  1 

58 

89  5 

58 

89  8 

59 

89  12 

59 

15 

88 

59 

89  16 

59 

89  19 

59 

21 

59 

23 

59 

26 

15  0 

28 

14  0 

30 

89 

20  0 

38 

19  0 

39 

18  0 

40 

17  0 

42 

16  0 

43 

0 

44 

0 

45 

90 

0 

90  0 

0 

90  0 

0 

90  0 

0 

90  0 

0 

90  0 

0 

90  0 

0 

90  0 

314 


Table  13,    Kelvin's  Sumner  Line  Table 


b 

a  = 

77" 

a  = 

=  78° 

a  = 

=  79° 

a  = 

=  80° 

a  = 

=  81° 

a  = 

■■  83° 

a  = 

=  83° 

K 

Q 

K 

Q 

K 

Q 

K 

Q 

K 

Q 

K 

Q 

K 

Q 

c 

o    / 

=  / 

o    / 

o    / 

0     / 

o    / 

o    / 

o    / 

0     / 

o    / 

o    / 

o    / 

o    / 

0    , 

0 

0  0 

77  0 

0  0 

78  0 

0  0 

79  0 

0  0 

80  0 

0  0 

81  0 

0  0 

82  0 

0  0 

83  0 

1 

14 

0 

12 

0 

J  11 

0 

10 

0 

9 

0 

8 

0 

7 

0 

2 

27 

0 

25 

0 

23 

0 

21 

0 

19 

0 

17 

0 

15 

0 

3 

41 

1 

37 

1 

34 

1 

31 

1 

28 

1 

25 

1 

22 

1 

4 

54 

2 

50 

2 

46 

2 

42 

1 

38 

1 

33 

1 

29 

1 

5 

1  7 

3 

1  2 

3 

57 

3 

52 

2 

47 

2 

42 

2 

37 

2 

6 

21 

4 

15 

4 

1  9 

4 

1  2 

3 

56 

3 

50 

3 

44 

2 

7 

34 

6 

27 

5 

20 

5 

13 

4 

1  6 

4 

58 

4 

51 

3 

8 

48 

7 

39 

7 

31 

6 

23 

6 

15 

5 

1  7 

5 

58 

4 

9 

2  1 

9 

52 

9 

43 

8 

33 

7 

24 

6 

15 

6 

1  6 

5 

10 

14 

11 

2  4 

11 

54 

10 

44 

9 

33 

8 

23 

7 

13 

6 

11 

28 

14 

17 

13 

2  5 

12 

54 

11 

43 

10 

31 

9 

20 

8 

12 

41 

16 

29 

15 

16 

14 

2  4 

13 

52 

12 

40 

10 

27 

9 

13 

54 

19 

41 

18 

28 

16 

14 

15 

2  1 

14 

48 

12 

34 

11 

14 

3  7 

22 

53 

21 

39 

19 

24 

17 

10 

16 

56 

14 

41 

12 

15 

20 

26 

3  5 

24 

50 

22 

34 

20 

19 

18 

2  4 

16 

48 

14 

IG 

33 

29 

17 

27 

3  1 

25 

44 

23 

28 

20 

12 

18 

55 

16 

17 

46 

33 

29 

30 

12 

28 

54 

26 

37 

23 

20 

21 

2  2 

18 

18 

59 

37 

41 

34 

23 

31 

3  4 

29 

46 

26 

28 

23 

9 

20 

19 

4  12 

41 

53 

38 

34 

35 

14 

32 

55 

29 

36 

26 

16 

23 

20 

25 

45 

4  5 

42 

45 

39 

24 

35 

3  4 

32 

44 

29 

23 

25 

21 

37 

50 

17 

46 

55 

43 

34 

39 

13 

35 

52 

32 

30 

28 

22 

50 

55 

28 

51 

4  6 

47 

44 

43 

22 

39 

59 

35 

37 

30 

23 

5  3 

78  0 

40 

56 

17 

51 

53 

47 

30 

42 

3  7 

38 

44 

33 

24 

15 

5 

51 

79  1 

27 

56 

4  3 

51 

39 

46 

15 

41 

50 

36 

25 

27 

11 

5  3 

6 

38 

80  1 

13 

55 

47 

50 

22 

44 

57 

39 

26 

39 

17 

14 

11 

48 

6 

22 

81  0 

56 

54 

30 

48 

3  4 

42 

27 

51 

23 

25 

16 

58 

11 

31 

4 

4  5 

58 

37 

52 

10 

45 

28 

6  3 

29 

36 

22 

5  8 

16 

41 

9 

13 

82  2 

45 

56 

17 

49 

29 

15 

35 

47 

28 

18 

21 

50 

14 

21 

7 

52 

83  0 

23 

52 

30 

27 

41 

58 

34 

28 

27 

59 

19 

29 

11 

59 

4 

30 

56 

31 

39 

48 

6  9 

40 

38 

33 

5  8 

24 

37 

16 

4  7 

8 

36 

84  0 

32 

51 

55 

20 

47 

48 

39 

17 

30 

45 

21 

14 

12 

42 

3 

33 

7  2 

79  3 

30 

54 

58 

45 

26 

35 

53 

26 

21 

17 

48 

7 

34 

14 

10 

41 

80  1 

6  8 

51 

34 

41 

5  1 

31 

28 

21 

54 

11 

35 

25 

17 

51 

8 

17 

57 

43 

47 

9 

36 

35 

26 

4  0 

15 

36 

36 

25 

7  1 

15 

27 

81  4 

52 

53 

17 

42 

42 

31 

6 

20 

37 

47 

33 

11 

22 

36 

11 

6  0 

59 

24 

47 

48 

36 

12 

24 

38 

58 

41 

21 

29 

45 

18 

8 

82  5 

32 

53 

55 

41 

18 

28 

39 

8  8 

50 

31 

37 

54 

25 

16 

12 

39 

59 

5  2 

46 

24 

33 

40 

19 

58 

41 

45 

7  3 

32 

24 

18 

46 

83  5 

8 

51 

30 

38 

41 

29 

80  7 

51 

53 

12 

39 

32 

25 

53 

11 

14 

57 

35 

42  ■ 

42 

39 

16 

8  0 

81  1 

20 

47 

40 

32 

6  0 

17 

21 

84  2 

41 

47 

43 

49 

25 

9 

10 

29 

55 

48 

39 

7 

23 

27 

8 

46 

52 

44 

59 

34 

18 

18 

37 

82  2 

56 

46 

14 

30 

33 

14 

52 

57 

45 

9  9 

43 

27 

27 

45 

10 

7  3 

54 

21 

37 

39 

20 

57 

85  2 

Table  13.    Kelvin's  Sumner  Line  Table 


315 


b 

a  =  77° 

a  =  78° 

a  =  79° 

a  =  80° 

a  =  81° 

a  =  83° 

a  =  83° 

K 

Q 

K 

Q 

K 

Q 

K 

Q 

K 

Q 

K 

Q 

K 

q: 

o 

o    / 

o  ,  , 

o    / 

o    / 

o    / 

o    / 

o    / 

o    / 

O    / 

o    / 

o   / 

O    / 

o    > 

45 

9  9 

80  43 

8  27 

81  27 

7  45 

82  10 

7  3 

82  54 

6  21 

83  37 

5  39 

84  20 

4  57 

85  2 

46 

.  19 

53 

36 

36 

53 

19 

11 

83  1 

28 

43 

45 

26 

5  2 

7 

47 

28 

81  3 

45 

45 

8  1 

27 

18 

9 

34 

50 

51 

32 

7 

13 

48 

37 

13 

53 

54 

9 

35 

25 

16 

41 

57 

56 

38 

12 

18 

49 

46 

23 

9  2 

82  4 

17 

44 

32 

24 

47 

84  4 

6  2 

44 

17 

24 

50 

55 

33 

10 

13 

24 

53 

39 

32 

53 

11 

7 

50 

21 

29 

51 

10  4 

44 

18 

23 

32 

83  2 

45 

40 

59 

18 

13 

57 

26 

35 

52 

13 

55 

26 

33 

39 

11 

52 

48 

7  5 

26 

18 

85  3 

31 

41 

53 

21 

82  6 

34 

43 

46 

20 

58 

57 

11 

33 

23 

10 

35 

46 

54 

29 

17 

41 

53 

53 

29 

8  4 

84  5 

16 

41 

28 

17 

40 

52 

55 

37 

28 

48 

83  3 

9  0 

38 

10 

13 

22 

49 

33 

23 

44 

58 

56 

45 

39 

55 

13 

6 

48 

16 

22 

27 

56 

38 

30 

48 

86  4 

57 

53 

50 

10  2 

24 

13 

57 

22 

31 

32 

85  4 

42 

37 

52 

10 

58 

11  0 

83  1 

9 

34 

19 

84  7 

28 

40 

37 

12 

47 

44 

56 

17 

59 

7 

13 

16 

45 

25 

17 

34 

49 

42 

20 

51 

52 

6  0 

23 

60 

14 

25 

23 

56 

31 

27 

39 

58 

47 

28 

55 

59 

4 

29 

61 

21 

37 

29 

84  7 

37 

37 

44 

85  7 

52 

37 

59 

86  6 

7 

36 

62 

28 

49 

35 

18 

42 

47 

49 

16 

57 

45 

7  3 

13 

11 

42 

63 

34 

84  1 

41 

29 

47 

57 

54 

25 

8  1 

53 

7 

21 

14 

49 

64 

40 

13 

46 

41 

52 

85  8 

59 

35 

5 

86  2 

11 

28 

17 

55 

65 

46 

25 

52 

52 

57 

18 

9  4 

44 

9 

10 

15 

36 

20 

87  2 

66 

52 

38 

57 

85  4 

10  2 

29 

8 

54 

13 

19 

19 

44 

23 

8 

67 

57 

51 

11  2 

15 

7 

39 

12 

86  4 

17 

28 

22 

51 

26 

15 

68 

12  2 

85  3 

7 

27 

11 

50 

16 

13 

21 

36 

25 

59 

29 

22 

69 

7 

16 

12 

39 

15 

86  1 

20 

23 

24 

45 

28 

87  7 

32 

29 

70 

12 

29 

16 

51 

19 

12 

24 

33 

27 

54 

31 

15 

35 

36 

71 

17 

42 

20 

86  3 

23 

23 

27 

43 

30 

87  3 

34 

23 

37 

43 

72 

21 

55 

24 

15 

27 

34 

30 

53 

33 

12 

37 

31 

39 

50 

73 

25 

86  8 

28 

27 

31 

45 

33 

87  3 

36 

21 

39 

39 

42 

57 

74 

29 

22 

32 

39 

34 

56 

36 

13 

39 

30 

41 

47 

44 

88  4 

75 

33 

35 

35 

51 

37 

87  7 

39 

23 

42 

39 

43 

55 

46 

11 

76 

37 

48 

38 

87  3 

40 

18 

42 

33 

44 

48 

45 

88  3 

48 

18 

77 

40 

87  2 

41 

16 

43 

30 

45 

44 

46 

58 

47 

11 

49 

25 

78 

43 

15 

44 

28 

46 

41 

47 

54 

48 

88  7 

49 

20 

51 

32 

79 

46 

29 

47 

41 

48 

53 

49 

88  4 

50 

16 

51 

28 

52 

39 

80 

48 

42 

49 

53 

50 

88  4 

51 

15 

52 

25 

53 

36 

54 

47 

81 

50 

56 

51 

88  6 

52 

15 

53 

25 

54 

35 

54 

44 

55 

54 

82 

52 

88  10 

53 

18 

54 

27 

54 

36 

55 

44 

55 

53 

56 

89  1 

83 

54 

23 

55 

31 

55 

39 

55 

46 

56 

54 

56 

89  1 

57 

9 

84 

56 

37 

56 

44 

56 

50 

56 

57 

57 

89  3 

57 

10 

58 

16 

85 

57 

51 

57 

56 

57 

89  2 

57 

89  7 

58 

13 

58 

18 

58 

23 

86 

58 

89  5 

58 

89  9 

58 

13 

58 

18 

59 

22 

59 

26 

59 

31 

87 

59 

18 

59 

22 

59 

25 

59 

28 

59 

31 

59 

35 

59 

38 

88 

13  0 

32 

12  0 

34 

11  0 

37 

10  0 

39 

9  0 

41 

8  0 

43 

7  0 

45 

89 

0 

46 

0 

47 

0 

48 

0 

49 

0 

50 

0 

52 

0 

53 

90 

0 

90  0 

0 

90  0 

0 

90  0 

0 

90  0 

0 

90  0 

0 

90  0 

0 

90  0 

316 


Table  13.    Kelyin's  Sumner  Line  Table 


b 

c 

a  = 

84° 

a  = 

85° 

a  = 

86° 

a  = 

87° 

a  = 

88° 

a  = 

89° 

a  = 

=  90° 

K 

Q 

K 

Q 

K 

Q 

K 

Q 

K 

Q 

K 

Q 

K 

Q 

o    / 

O     / 

o    / 

o    / 

o    / 

o    / 

O     1 

o    / 

o    / 

o    / 

o    / 

o   / 

0     / 

o    / 

0 

0  0 

84  0 

0  0 

85  0 

0  0 

86  0 

0  0 

87  0 

0  0 

88  0 

0  0 

89  0 

0  0 

90  0 

1 

6 

0 

5 

0 

4 

0 

3 

0 

2 

0 

1 

0 

0 

.  0 

2 

13 

0 

10 

0 

8 

0 

6 

0 

4 

0 

2 

0 

0 

0 

3 

19 

0 

16 

0 

13 

0 

9 

0 

6 

0 

3 

0 

0 

0 

4 

25 

1 

21 

1 

17 

1 

13 

0 

8 

0 

4 

0 

0 

0 

5 

31 

1 

26 

1 

21 

1 

16 

1 

10 

0 

5 

0 

0 

0 

6 

38 

2 

31 

2 

25 

1 

19 

1 

13 

1 

6 

0 

0 

0 

7 

44 

3 

37 

2 

29 

2 

22 

1 

15 

1 

7 

0 

0 

0 

8 

50 

3 

42 

3 

33 

2 

25 

2 

17 

1 

8 

1 

0 

0 

9 

56 

4 

47 

4 

38 

3 

28 

2 

19 

1 

9 

1 

0 

0 

10 

1  2 

5 

52 

5 

42 

4 

31 

3 

21 

2 

10 

1 

0 

0 

11 

9 

7 

57 

6 

46 

4 

34 

3 

23 

2 

11 

1 

0 

0 

12 

15 

8 

1  2 

7 

50 

5 

37 

4 

25 

3 

13 

1 

0 

0 

13 

21 

9 

7 

8 

54 

6 

40 

5 

27 

3 

14 

2 

0 

0 

14 

27 

11 

12 

9 

58 

7 

44 

5 

29 

4 

15 

2 

0 

0 

15 

33 

12 

18 

10 

1  2 

8 

47 

6 

31 

4 

16 

2 

0 

0 

16 

39 

14 

23 

12 

6 

9 

50 

7 

33 

5 

17 

2 

0 

0 

17 

45 

16 

28 

13 

10 

10 

53 

8 

35 

5 

18 

3 

0 

0 

18 

51 

18 

33 

15 

14 

12 

56 

9 

37 

6 

19 

3 

0 

0 

19 

57 

20 

38 

16 

18 

13 

59 

10 

39 

7 

20 

4 

0 

0 

20 

2  3 

22 

43 

18 

22 

14 

1  2 

11 

41 

7 

21 

4 

0 

0 

21 

9 

24 

48 

20 

26 

16 

4 

12 

43 

8 

22 

4 

0 

0 

22 

15 

26 

52 

22 

30 

17 

7 

13 

45 

9 

22 

4 

0 

0 

23 

20 

28 

57 

24 

34 

19 

10 

14 

47 

10 

23 

5 

0 

0 

24 

26 

31 

2  2 

26 

38 

21 

13 

16 

49 

10 

24 

5 

0 

0 

25 

32 

34 

7 

28 

41 

22 

16 

17 

51 

11 

25 

6 

0 

0 

26 

38 

36 

11 

30 

45 

24 

19 

18 

53 

12 

26 

6 

0 

0 

27 

43 

39 

16 

33 

49 

26 

22 

20 

54 

13 

27 

7 

0 

0 

28 

49 

42 

21 

35 

53 

28 

24 

21 

56 

14 

28 

7 

0 

0 

29 

54 

45 

25 

37 

56 

30 

27 

23 

58 

15 

29 

8 

0 

0 

30 

3  0 

48 

30 

40 

2  0 

32 

30 

24 

1  0 

16 

30 

8 

0 

0 

31 

5 

51 

35 

43 

4 

34 

33 

26 

2 

17 

31 

9 

0 

0 

32 

11 

54 

39 

45 

7 

36 

35 

27 

4 

18 

32 

9 

0 

0 

33 

16 

58 

43 

48 

11 

39 

38 

29 

5 

19 

33 

10 

0 

0 

34 

21 

85  1 

48 

51 

14 

41 

41 

31 

7 

20 

34 

10 

0 

0 

35 

26 

5 

52 

54 

18 

.43 

43 

33 

9 

22 

34 

11 

0 

0 

36 

31 

8 

56 

57 

21 

46 

46 

34 

11 

23 

35 

11 

0 

0 

37 

36 

12 

3  0 

86  0 

24 

48 

48 

36 

12 

24 

36 

12 

0 

0 

38 

41 

16 

5 

3 

28 

51 

51 

38 

14 

25 

37 

13 

0 

0 

39 

46 

20 

9 

7 

31 

53 

53 

40 

16 

27 

38 

14 

0 

0 

40 

51 

24 

13 

10 

34 

56 

56 

42 

17 

28 

39 

14 

0 

0 

41 

56 

28 

17 

13 

37 

59 

58 

44 

19 

29 

39 

15 

0 

0 

42 

4  1 

32 

21 

17 

41 

87  2 

2  0 

46 

20 

31 

40 

15 

0 

0 

43 

5 

36 

25 

20 

44 

4 

3 

48 

22 

32 

41 

16 

0 

0 

44 

10 

41 

28 

24 

47 

7 

5 

50 

23 

34 

42 

17 

0 

0 

45 

14 

45 

32 

28 

50 

10 

7 

53 

25 

35 

43 

18 

0 

0 

Table  13.    Kelvin's  Sumner  Line  Table 


317 


b 

o 

a  =  84° 

a  =  85° 

a  =  86° 

a  = 

87° 

a  = 

88° 

a  = 

89° 

a  = 

90° 

K 

Q 

K 

Q 

K 

Q 

K 

Q 

K 

Q 

K 

Q 

K 

Q 

o   / 

o    / 

o    / 

o    / 

o    / 

o    / 

0    / 

o    / 

°     f 

o    / 

o    / 

o    / 

o    / 

o    / 

45 

4  14 

85  45 

3  32 

86  28 

2  50 

87  10 

2  7 

87  53 

I  25 

88  35 

0  43 

89  18 

0  0 

90  0 

46 

19 

49 

36 

31 

53 

13 

9 

55 

26 

37 

43 

18 

0 

0 

47 

23 

54 

39 

35 

55 

16 

12 

57 

28 

38 

44 

19 

0 

0 

48 

27 

59 

43 

39 

58 

19 

14 

59 

29 

40 

45 

20 

0 

0 

49 

i 

31 

86  3 

46 

43 

3  1 

22 

16 

88  2 

31 

41 

45 

21 

0 

0 

50 

35 

8 

50 

47 

4 

26 

18 

4 

32 

43 

46 

21 

0 

0 

51 

39 

13 

53 

51 

7 

29 

20 

7 

33 

44 

46 

22 

0 

0 

52 

43 

18 

56 

55 

9 

32 

22 

9 

35 

46 

47 

23 

0 

0 

53 

47 

23 

59 

59 

12 

35 

24 

12 

36 

48 

48 

24 

0 

0 

54 

51 

28 

4  2 

87  3 

14 

39 

26 

14 

37 

49 

49 

25 

0 

0 

55 

55 

33 

5 

8 

17 

42 

27 

17 

38 

51 

49 

26 

0 

0 

56 

58 

38 

8 

12 

19 

46 

29 

19 

39 

53 

50 

26 

0 

0 

57 

5  2 

43 

11 

16 

21 

49 

31 

22 

41 

55 

50 

27 

0 

0 

58 

5 

49 

14 

21 

24 

53 

33 

25 

42 

56 

51 

28 

0 

0 

59 

8 

54 

17 

25 

26 

56 

34 

27 

43 

58 

51 

29 

0 

0 

60 

11 

87  0 

20 

30 

28 

88  0 

36 

30 

44 

89  0 

52 

30 

0 

0 

61 

14 

5 

22 

34 

30 

3 

37 

33 

45 

2 

52 

31 

0 

0 

62 

17 

11 

25 

39 

32 

7 

39 

35 

46 

4 

53 

32 

0 

0 

63 

20 

16 

27 

44 

34 

11 

40 

38 

47 

5 

53 

33 

0 

0 

64 

23 

22 

30 

48 

36 

15 

42 

41 

48 

7 

54 

34 

■   0 

0 

65 

26 

27 

32 

53 

38 

18 

43 

44 

49 

9 

54 

35 

0 

0 

66 

29 

33 

34 

58 

39 

22 

44 

47 

50 

11 

55 

36 

0 

0 

67 

31 

39 

36 

88  3 

41 

26 

46 

50 

50 

13 

55 

37 

0 

0 

68 

34 

45 

38 

7 

42 

30 

47 

53 

51 

15 

56 

38 

0 

0 

69 

36 

51 

40 

12 

44 

34 

48 

55 

52 

17 

56 

38 

0 

0 

70 

38 

56 

42 

17 

46 

38 

49 

58 

53 

3 

19 

56 

39 

0 

0 

71 

40 

88  2 

44 

22 

47 

42 

50 

89  1 

53 

21 

56 

40 

0 

0 

72 

42 

8 

45 

27 

48 

46 

51 

4 

54 

23 

57 

41 

0 

0 

73 

44 

14 

47 

32 

50 

50 

52 

7 

55 

25 

57 

42 

0 

0 

74 

46 

20 

49 

37 

51 

54 

53 

10 

55 

27 

58 

43 

0 

0 

75 

48 

27 

50 

42 

52 

58 

54 

13 

56 

29 

58 

44 

0 

0 

76 

49 

33 

51 

47 

53 

89  2 

55 

16 

56 

31 

58 

45 

0 

0 

77 

51 

39 

52 

52 

54 

6 

55 

19 

57 

33 

58 

46 

0 

0 

78 

52 

45 

53 

57 

55 

10 

56 

23 

57 

35 

59 

47 

0 

0 

79 

53 

51 

54 

89  3 

56 

14 

57 

26 

58 

37 

59 

49 

0 

0 

80 

54 

57 

55 

8 

56 

18 

57 

29 

58 

39 

59 

50 

0 

0 

81 

55 

89  3 

56 

13 

57 

22 

58 

32 

59 

41 

59 

51 

0 

0 

82 

56 

10 

57 

18 

58 

27 

58 

35 

59 

43 

59 

52 

0 

0 

83 

57 

16 

58 

23 

58 

31 

59 

38 

59 

45 

1  0 

53 

0 

0 

84 

58 

22 

58 

29 

59 

35 

59 

41 

59 

47 

0 

54 

0 

0 

85 

59 

29 

59 

34 

59 

39 

59 

44 

2  0 

50 

0 

55 

0 

0 

86 

59 

35 

59 

39 

59 

43 

3  0 

47 

0 

52 

0 

56 

0 

0 

87 

6  0 

41 

5  0 

44 

4  0 

47 

0 

51 

0 

54 

0 

57 

0 

0 

88 

0 

47 

0 

50 

0 

52 

0 

54 

0 

56 

0 

58 

0 

0 

89 

0 

54 

0 

55 

0 

56 

0 

57 

0 

58 

0 

59 

0 

0 

90 

0 

90  0 

0 

90  0 

0 

90  0 

0 

90  0 

0 

90  0 

0 

90  0 

0 

0 

318 


Table  14.    Sumner  Intersection 


Larger 
Bearing 


34^ 

36 

38 

40 

42 

44 

46 

48 

50 

52 

54 

56 

58 

60 

62 

64 

66 

68 

70 

72 

74 

76 

78 

80 

82 

84 

86 

88 

90 

92 

94 

96 

98 

100 

102 

104 

106 

108 

110 

112 

114 

116 

118 

120 

122 

124 

126 

128 

130 

132 

134 

136 

138 

140 

142 

144 

146 

148 

150 

152 

154 

156 

158 

160 


Smaller  Bearing 


0.07 
.06 
.06 
.06 
.05 
.05 
.05 
.05 
.05 
.05 
.04 
.04 
.04 
.04 
-.04 
.04 
.04 
.04 
.04 
.04 
.04 
.04 
.04 
.04 
.04 
.04 
.04 
.04 
.03 
.03 
.03 
.03 
.04 
.04 
.04 
.04 
.04 
.04 
.04 
.04 
.04 
.04 
.04 
.04 
.04 
.04 
.04 
.04 
.04 
.05 
.05 
.05 
.05 
.05 
.05 
.06 
.06 
.06 
.07 
.07 
.07 
.08 
.09 
.09 


0.14 
.13 
.12 
.12 
.11 
.11 
.10 
.10 
.10 
.09 
.09 
.09 
.09 
.08 
.08 
.08 
.08 
.08 
.08 
.08 
.07 
.07 
.07 
.07 
.07 
.07 
.07 
.07 
.07 
.07 
.07 
.07 
.07 
.07 
.07 
.07 
.07 
.07 
.07 
.07 
.07 
.07 
.08 
.08 
.08 
.08 
.08 
.08 
.09 
.09 
.09 
.09 
.10 
.10 
.10 
.11 
.12 
.12 
.12 
.13 
.14 
.15 
.16 
.17 


0.22 
.21 
.20 
.19 
.18 
.17 
.16 
.16 
.15 
.15 
.14 
.14 
.13 
.13 
.13 
.12 
.12 
.12 
.12 
.11 
.11 
.11 
.11 
.11 
.11 
.11 
.11 
.11 
.11 
.10 
.10 
.10 
.10 
.11 
.11 
.11 
.11 
.11 
.11 
.11 
.11 
.11 
.11 
.11 
.12 
.12 
.12 
.12 
.13 
.13 
.13 
.14 
.14 
.15 
.15 
.16 
.16 
.17 
.18 
.19 
.20 
.21 
.22 
.24 


8°   10' 


0.32 
.30 
.28 
.26 
.25 
.24 
.23 
.22 
.21 
.20 
.19 
.19 
.18 
.18 
.17 
.17 
.16 
.16 
.16 
.15 
.15 
.15 
.15 
.15 
.14 
.14 
.14 
.14 
.14 
.14 
.14 
.14 
.14 
.14 
.14 
.14 
.14 
.14 
.14 
.14 
.15 
.15 
.15 
.15 
.15 
.16 
.16 
.16 
.17 
.17 
.17 
.18 
.18 
.19 
.19 
.20 
.21 
.22 
.23 
.24 
.25 
.26 
.28 
.30 


0.43 
.40 
.37 
.35 
.33 
.31 
.30 
.28 
.27 
.26 
.25 
.24 
.23 
.23 
.22 
.21 
.21 
.20 
.20 
.20 
.19 
.19 
.19 
.18 
.18 
.18 
.18 
.18 
.18 
.18 
.17 
.17 
.17 
.17 
.17 
.17 
.17 
.18 
.18 
.18 
.18 
.18 
.18 
.18 
.19 
.19 
.19 
.20 
.20 
.20 
.21 
.21 
.22 
.23 
.23 
.24 
.25 
.26 
.27 
•28 
.30 
.31 
.33 
.35 


Smaller  Bearing 

^ 

Larger 
Bearing 

12° 

14° 

16° 

18" 

20° 

42° 

0.42 

0.52 

0.63 

0.76 

0.91 

44 

.39 

.48 

.59 

.70 

.84 

46 

.37 

.46 

.55 

.66 

.78 

48 

.35 

.43 

.52 

.62 

.73 

50 

.34 

.41 

.49 

.58 

.68 

52 

.32 

.39 

.47 

.55 

.65 

54 

.31 

.38 

.45 

.52 

.61 

56 

.30 

.36 

.43 

.50 

.58 

58 

.29 

.35 

.41 

.48 

.56 

60 

.28 

.34 

.40 

.46 

.53 

62 

.27 

.33 

.38 

.44 

.51 

64 

.26 

.32 

.37 

.43 

.49 

66 

.26 

.31 

.36 

.42 

.48 

68 

.25 

.30 

.35 

.40 

.46 

70 

.24 

.29 

.34 

.39 

.45 

72 

.24 

.28 

.33 

.38 

.43 

74 

.23 

.28 

.32 

.37 

.42 

76 

.23 

.27 

.32 

.36 

.41 

78 

.23 

.27 

.31 

.36 

.40 

80 

.22 

.26 

.31 

.35 

.39 

82 

.22 

.26 

.30 

.34 

.39 

84 

.22 

.26 

.30 

.34 

.38 

86 

.22 

.25 

.29 

.33 

.37 

88 

.21 

.25 

.29 

.33 

.37 

90 

.21 

.25 

.29 

.32 

.36 

92 

.21 

.25 

.28 

.32 

.36 

94 

.21 

.25 

.28 

.32 

.36 

96 

.21 

.24 

.28 

.32 

.35 

98 

.21 

.24 

.28 

.31 

.35 

100 

.21 

.24 

.28 

.31 

.35 

102 

.21 

.24 

.28 

.31 

.35 

104 

.21 

.24 

.28 

.31 

.34 

106 

.21 

.24 

.28 

.31 

.34 

108 

.21 

.24 

.28 

.31 

.34 

110 

.21 

.24 

.28 

.31 

.34 

112 

.21 

.24 

.28 

.31 

.34 

114 

.21 

.24 

.28 

.31 

.34 

116 

.21 

.25 

.28 

.31 

.34 

118 

.22 

.25 

.28 

.31 

.35 

120 

.22 

.25 

.28 

.32 

.35 

122 

.22 

.25 

.29 

.32 

.35 

124 

.22 

.26 

.29 

.32 

.35 

126 

.23 

.26 

.29 

.32 

.36 

128 

.23 

.26 

.30 

.33 

.36 

130 

.23 

.27 

.30 

.33 

.36 

132 

.24 

.27 

.31 

.34 

.37 

134 

.24 

.28 

.31 

.34 

.37 

136 

.25 

.28 

.32 

.35 

.38 

138 

.26 

.29 

.32 

.36 

.39 

140 

.26 

.30 

.33 

.36 

.39 

142 

.27 

.31 

.34 

.37 

.40 

144 

.28 

.32 

.35 

.38 

.41 

146 

.29 

.33 

.36 

.39 

.42 

148 

.30 

.34 

.37 

.40 

.43 

150 

.31 

.35 

.38 

.42 

.45 

152 

.32 

.36 

.40 

.43 

.46 

154 

.34 

.38 

.41 

.44 

.48 

156 

.35 

.39 

.43 

.46 

.49 

158 

.37 

.41 

.45 

.48 

.51 

160 

.39 

.43 

.47 

.50 

.53 

162 

.41 

.46 

.49 

.52 

.56 

164 

.44 

.48 

.52 

.55 

.58 

166 

.47 

.52 

.55 

.58 

.61 

168 

.50 

.55 

.59 

.62 

.65 

Table  14.    Sumner  Intersection 


319 


Larger 
Bearing 


54° 

56 

58 

60 

62 

64 

66 

68 

70 

72 

74 

76 

78 

80 

82 

84 

86 

88 

90 

92 

94 

96 

98 

100 

102 

104 

106 

108 

110 

112 

114 

116 

118 

120 

122 

124 

126 

128 

130 

132 

134 

136 

138 

140 

142 

144 

146 

148 

150 

152 

154 

156 

158 

160 

162 

164 

166 

168 

170 

172 

174 

176 

178 


Smaller  Bearing 


22°  24°  26°  28 


0.71 
.67 
.64 
.61 
.58 
.56 
.54 
.52 
.50 
.49 
.48 
.46 
.45 
.44 
.43 
.42 
.42 
.41 
.40 
.40 
.39 
.39 
.39 
.38 
.38 
.38 
.38 
.38 
.37 
.37 
.37 
.38 
.38 
.38 
.38 
.38 
.39 
.39 
.39 
.40 
.40 
.41 
.42 
.42 
.43 
.44 
.45 
.46 
.48 
.49 
.50 
.52 
.54 
.56 
.58 
.61 
.64 
.67 
.71 
.75 
.80 
.85 
.92 


0.81 
.77 
.73 
.69 
.66 
.63 
.61 
.59 
.57 
.55 
.53 
.52 
.50 
.49 
.48 
.47 
.46 
.45 
.45 
.44 
.43 
.43 
.42 
.42 
.42 
.41 
.41 
.41 
.41 
.41 
.41 
.41 
.41 
.41 
.41 
.41 
.42 
.42 
.42 
.43 
.43 
.44 
.45 
.45 
.46 
.47 
.48 
.49 
.50 
.52 
.53 
.55 
.57 
.59 
.61 
.63 
.66 
.69 
.73 
.77 
.81 
.87 
.93 


0.93 
.88 
.83 
.78 
.75 
.71 
.68 
.66 
.63 
.61 
.59 
.57 
.56 
.54 
.53 
.52 
.51 
.50 
.49" 
.48 
.47 
.47 
.46 
.46 
.45 
.45 
.45 
.44 
.44 
.44 
.44 
.44 
.44 
.44 
.44 
.44 
.45 
.45 
.45 
.46 
.46 
.47 
.47 
.48 
.49 
.50 
.51 
.52 
.53 
.54 
.56 
.57 
.59 
.61 
.63 
.66 
.68 
.71 
.75 
.78 
.83 
.88 
.93 


1.07 
1.00 
0.94 
.89 
.84 
.80 
.76 
.73 
.70 
.68 
.65 
.63 
.61 
.60 
.58 
.57 
.55 
.54 
.53 
.52 
.51 
.51 
.50 
.49 
.49 
.48 
.48 
.48 
.47 
.47 
.47 
.47 
.47 
.47 
.47 
.47 
.47 
.48 
.48 
.48 
.49 
.49 
.50 
.51 
.51 
.52 
.53 
.54 
.55 
.57 
.58 
.60 
.61 
.63 
.65 
.68 
.70 
.73 
.76 
.80 
.84 
.89 
.94 


30' 


1.23 
1.14 
1.07 
1.00 
0.94 
.89 
.85 
.81 
.78 
.75 
.72 
.70 
.67 
.65 
.63 
.62 
.60 
.59 
.58 
.57 
.56 
.55 
.54 
.53 
.53 
.52 
.52 
.51 
.51 
.50 
.50 
.50 
.50 
.50 
.50 
.50 
.50 
.50 
.51 
.51 
.52 
.52 
.53 
.53 
.54 
.55 
.56 
.57 
.58 
.59 
.60 
.62 
.63 
.65 
.67 
.70 
.72 
.75 
.78 
.81 
.85 
.89 
.94 


Smaller  Bearing 

Larger 
Bearing 

32° 

34° 

36° 

38° 

40° 

62° 

1.06 

1.19 

1.34 

1.51 

1.72 

64 

1.00 

1.12 

1.25 

1.40 

1.58 

66 

0.95 

1.06 

1.18 

1.31 

1.47 

68 

.90 

1.00 

1.11 

1.23 

1.37 

70 

.86 

0.95 

1.05 

1.16 

1.29 

72 

.82 

.91 

1.00 

1.10 

1.21 

74 

.79 

.87 

0.95 

1.05 

1.15 

76 

.76 

.84 

.91 

1.00 

1.09 

78 

.74 

.80 

.88 

0.96 

1.04 

80 

.71 

.78 

.85 

.92 

1.00 

82 

.69 

.75 

.82 

.89 

0.96 

84 

.67 

.73 

.79 

.86 

.93 

86 

.66 

.71 

.77 

.83 

.89 

88 

.64 

.69 

.75 

.80 

.86 

90 

.62 

.67 

.73 

.78 

.84 

92 

.61 

.66 

.71 

.76 

.82 

94 

.60 

.65 

.69 

.74 

.79 

96 

.59 

.63 

.68 

.73 

.78 

98 

.58 

.62 

.67 

.71 

.76 

100 

.57 

.61 

.65 

.70 

.74 

102 

.56 

.60 

.64 

.68 

.73 

104 

.56 

.60 

.63 

.67 

.72 

106 

.55 

.59 

.63 

.66 

.70 

108 

.55 

.58 

.62 

.66 

.69 

110 

.54 

.58 

.61 

.65 

.68 

112 

.54 

.57 

.61 

.64 

.68 

114 

.54 

.57 

.60 

.63 

.67 

116 

.53 

.56 

.60 

.63 

.66 

118 

.53 

.56 

.59 

.63 

.66 

120 

.53 

.56 

.59 

.62 

.65 

122 

.53 

.56 

.59 

.62 

.65 

124 

.53 

.56 

.59 

.62 

.65 

126 

.53 

.56 

.59 

.62 

.64 

128 

.53 

.56 

.59 

.62 

.64 

130 

.54 

.56 

.59 

.62 

.64 

132 

.54 

.56 

.59 

.62 

.64 

134 

.54 

.57 

.59 

.62 

.64 

136 

.55 

.57 

.60 

.62 

.65 

138 

.55 

.58 

.60 

.63 

.65 

140 

.56 

.58 

.61 

.63 

.65 

142 

.56 

.59 

.61 

.63 

.66 

144 

.57 

.60 

.62 

.64 

.66 

146 

.58 

.60 

.63 

.65 

.67 

148 

.59 

.61 

.63 

.66 

.68 

150 

.60 

.62 

.64 

.66 

.68 

152 

.61 

.63 

.65 

.67 

.69 

154 

.62 

.65 

.67 

.68 

.70 

156 

.64 

.66 

.68 

.70 

.72 

158 

.66 

.67 

.69 

.71 

.73 

160 

.67 

.69 

.71 

.73 

.74 

162 

.69 

.71 

.73 

.74 

.76 

164 

.71 

.73 

.75 

.76 

.78 

166 

.74 

.75 

.77 

.78 

.79 

168 

.76 

.78 

.79 

.80 

.82 

170 

.79 

.80 

.82 

.83 

84 

172 

.82 

.84 

.85 

.86 

.86 

174 

.86 

.87 

.88 

.89 

.89 

176 

.90 

.91 

.91 

.92 

.93 

178 

.95 

.95 

.95 

.96 

.96 

320 


Table  14.    Sumner  Intersection 


Smaller  Bearing 

Smaller  Bearing      I 

Lakger 
Bearing 

Larger 
Bearing 

1 

42° 

44° 

46° 

48° 

50° 

52° 

54° 

56° 

58° 

60° 

72° 

1.34 

1.48 

1.64 

1.83 

2.04 

74 

1.26 

1.39 

1.53 

1.70 

1.88 

76 

1.20 

1.31 

1.44 

1.58 

1.75 

78 

1.14 

1.24 

1.36 

1.49 

1.63 

80 

1.09 

1.18 

1.28 

1.40 

1.53 

82 

1.04 

1.13 

1.22 

1.33 

1.45 

82° 

1.58 

1.72 

1.89 

2.08 

2.31 

84 

1.00 

1.08 

1.17 

1.26 

1.37 

84 

1.49 

1.62 

1.77 

1.93 

2.13 

86  ■ 

0.96 

1.04 

1.12 

1.21 

1.30 

86 

1.41 

1.53 

1.66 

1.81 

1.98 

88 

.93 

1.00 

1.08 

1.16 

1.24 

88 

1.34 

1.45 

1.56 

1.70 

1.84 

90 

.90 

0.97 

1.04 

1.11 

1.19 

90 

1.28 

1.38 

1.48 

1.60 

1.73 

92 

.87 

.93 

1.00 

1.07 

1.14 

92 

1.23 

1.31 

1.41 

1.52 

1.63 

94 

.85 

.91 

0.97 

1.03 

1.10 

94 

1.18 

1.26 

1.35 

1.44 

1.55 

96 

.83 

.88 

.94 

1.00 

1.06 

96 

1.13 

1.21 

1.29 

1.38 

1.47 

98 

.81 

.86 

.91 

0.97 

1.03 

98 

1.10 

1.16 

1.24 

1.32 

1.41 

100 

.79 

.84 

.89 

.94 

1.00 

100 

1.06 

1.12 

1.19 

1.27 

1.35 

102 

.77 

.82 

.87 

.92 

0.97 

102 

1.03 

1.09 

1.15 

1.22 

1.29 

104 

.76 

.80 

.85 

.90 

.95 

104 

1.00 

1.06 

1.12 

1.18 

1.25 

106 

.74 

.79 

.83 

.88 

.92 

106 

0.97 

1.03 

1.09 

1.14 

1.20 

108 

.73 

.77 

.81 

.86 

.90 

108 

.95 

1.00 

1.05 

1.11 

1.17 

110 

.72 

.76 

.80 

.84 

.88 

110 

.93 

0.98 

1.02 

1.08 

1.13 

112 

.71 

.75 

.79 

.83 

.87 

112 

.91 

.95 

1.00 

1.05 

1.10 

114 

.70 

.74 

.78 

.81 

.85 

114 

.89 

.93 

0.98 

1.02 

1.07 

116 

.70 

.73 

.77 

.80 

.84 

116 

.88 

.92 

.96 

1.00 

1.04 

118 

.69 

.72 

.76 

.79 

.83 

118 

.86 

.90 

.94 

0.98 

1.02 

120 

.68 

.72 

.75 

.78 

.82 

120 

.85 

.89 

.91 

.96 

1.00 

122 

.68 

.71 

.74 

.77 

.81 

122 

.84 

.87 

.90 

.95 

0.98 

124 

.68 

.71 

.74 

.77 

.80 

124 

.83 

.86 

.90 

.93 

.96 

126 

.67 

.70 

.73 

.76 

.79 

126 

.82 

.85 

.88 

.91 

.95 

128 

.67 

.70 

.73 

.75 

.78 

128 

.81 

.84 

.87 

.90 

.93 

130 

.67 

.70 

.72 

.75 

.78 

130 

.81 

.83 

.86 

.89 

.92 

132 

.67 

.70 

.72 

.75 

.77 

132 

.80 

.83. 

.85 

.88 

.91 

134 

.67 

.69 

.72 

.74 

.77 

134 

.80 

.82 

.85 

.87 

.90 

136 

.67 

.70 

.72 

.74 

.77 

136 

.80 

.82 

.84 

.87 

.89 

138 

.67 

.70 

.72 

.74 

.77 

138 

.79 

.81 

.84 

.86 

.89 

140 

.68 

.70 

.72 

.74 

.77 

140 

.79 

.81 

.83 

.86 

.88 

142 

.68 

.70 

.72 

.74 

.77 

142 

.79 

.81 

.83 

.85 

.87 

144 

.68 

.71 

.73 

.75 

.77 

144 

.79 

.81 

.83 

.85 

.87 

146 

.69 

.71 

.73 

.75 

.77 

146 

.79 

.81 

.83 

.85 

.87 

148 

.70 

.72 

.74 

.76 

.77 

148 

.79 

.81 

.83 

.85 

.87 

150 

.70 

.72 

.74 

.76 

.78 

150 

.80 

.81 

.83 

.85 

.87 

152 

.71 

.73 

.75 

.77 

.78 

152 

.80 

.82 

.83 

.85 

.87 

154 

.72 

.74 

.76 

.77 

.79 

154 

.81 

.82 

.84 

.85 

.87 

156 

.73 

.75 

.77 

.78 

.80 

156 

.81 

.83 

.84 

.86 

.87 

158 

.74 

.76 

.78 

.79 

.81 

158 

.82 

.83 

.85 

.86 

.87 

160 

.76 

.77 

.79 

.80 

.82 

160 

.83 

.84 

.85 

.86 

.88 

162 

.77 

.79 

.80 

.81 

.83 

162 

.84 

.85 

.86 

.87 

.89 

164 

.79 

.80 

.81 

.83 

.84 

164 

.85 

.86 

.87 

.88 

.89 

166 

.81 

.82 

.83 

.84 

.85 

166 

.86 

.87 

.88 

.89 

.90 

168 

.83 

.84 

.85 

.86 

.87 

168 

.88 

.89 

.90 

.90 

.91 

170 

.85 

.86 

.87 

.88 

.88 

170 

.89 

.90 

.90 

.91 

.92 

172 

.87 

.88 

.89 

.90 

.90 

172 

.91 

.92 

.91 

.93 

.93 

174 

.90 

.91 

.91 

.92 

.92 

174 

.93 

.93 

.94 

.95 

.95 

176 

.93 

.93 

.94 

.94 

.95 

176 

.95 

.95 

.96 

.96 

.96 

178 

.96 

.97 

.97 

.97 

.97 

178 

.97 

.98 

.98 

.98 

.98 

Table  14.    Sumner  Intersection 


321 


Smaller  Bearing 

Smaller  Bearing     I 

Larger 
Bearing 

Larger 
Bearing 

1 

62° 

64° 

66° 

68° 

70° 

72° 

74° 

76° 

78° 

80° 

92° 

1.77 

1.91 

2.08 

2.28 

2.&1 

• 

94 

1.67 

1.80 

1.95 

2.12 

2.31 

96 

1.58 

1.70 

1.83 

1.97 

2.14 

98 

1.50 

1.61 

1.72 

1.85 

2.00 

100 

1.43 

1.53 

1.63 

1.75 

1.88 

102 

1.37 

1.46 

1.55 

1.66 

1.77 

102° 

1.90 

2.05 

2.21 

2.40 

2.63 

104 

1.32 

1.40 

1.48 

1.58 

1.68 

104 

1.79 

1.92 

2.07 

2.23 

2.42 

106 

1.27 

1.34 

1.42 

1.51 

1.60 

106 

1.70 

1.81 

1.94 

2.08 

2.25 

108 

1.23 

1.29 

1.37 

1.44 

1.53 

108 

1.62 

1.72 

1.83 

1.96 

2.10 

110 

1.19 

1.25 

1.32 

1.39 

1.46 

110 

1.54 

1.64 

1.74 

1.85 

1.97 

112 

1.15 

1.21 

1.27 

1.33 

1.40 

112 

1.48 

1.56 

1.65 

1.75 

1.86 

114 

1.12 

1.17 

1.23 

1.29 

1.35 

114 

1.42 

1.50 

1.58 

1.66 

1.76 

116 

1.09 

1.14 

1.19 

1.25 

1.31 

116 

1.37 

1.44 

1.51 

1.59 

1.68 

118 

1.07 

1.11 

1.16 

1.21 

1.26 

118 

1.32 

1.38 

1.45 

1.52 

1.60 

120 

1.04 

1.08 

1.13 

1.18 

1.23 

120 

1.28 

1.34 

1.40 

1.46 

1.53 

122 

1.02 

1.06 

1.10 

1.15 

1.19 

122 

1.24 

1.29 

1.35 

1.41 

1.47 

124 

1.00 

1.04 

1.08 

1.12 

1.16 

124 

1.21 

1.25 

1.31 

1.36 

1.42 

126 

0.98 

1.02 

1.05 

1.09 

1.13 

126 

1.18 

1.22 

1.27 

1.32 

1.37 

128 

.97 

1.00 

1.03 

1.07 

1.11 

128 

1.15 

1.19 

1.23 

1.28 

1.33 

130 

.95 

0.98 

1.02 

1.05 

1.09 

130 

1.12 

1.16 

1.20 

1.24 

1.29 

132 

.94 

.97 

1.00 

1.03 

1.06 

132 

1.10 

1.13 

1.17 

1.21 

1.25 

134 

.93 

.96 

0.99 

1.01 

1.04 

134 

1.08 

1.11 

1.14 

1.18 

1.22 

136 

.92 

.95 

.97 

1.00 

1.03 

136 

1.06 

1.09 

1.12 

1.15 

1.19 

138 

.91 

.94 

.96 

0.99 

1.01 

138 

1.04 

1.07 

1.10 

1.13 

1.16 

140 

.90 

.93 

.95 

.97 

1.00 

140 

1.03 

1.05 

1.08 

1.11 

1.14 

142 

.90 

.92 

.94 

.96 

0.99 

142 

1.01 

1.04 

1.06 

1.09 

1.12 

144 

.89 

.91 

.93 

.96 

.98 

144 

1.00 

1.02 

1.05 

1.07 

1.10 

146 

.89 

.91 

.93 

.95 

.97 

146 

0.99 

1.01 

1.03 

1.05 

1.08 

148 

.89 

.90 

.92 

.94 

.96 

148; 

.98 

1.00 

1.02 

1.04 

1.06 

150 

.88 

.90 

.92 

.94 

.95 

150 

.97 

0.99 

1.01 

1.03 

1.05 

152 

.88 

.90 

.92 

.93 

.95 

152 

.97 

.98 

1.00 

1.02 

1.04 

154 

.88 

.90 

.91 

.93 

.94 

154 

.96 

.98 

0.99 

1.01 

1.02 

156 

.89 

.90 

.91 

.93 

.94 

156 

.96 

.97 

.99 

1.00 

1.01 

158 

.89 

.90 

.91 

.93 

.94 

158 

.95 

.97 

.98 

0.99 

1.01 

160 

.89 

.90 

.91 

.93 

.94 

160 

.95 

.96 

.98 

.99 

1.00 

162 

.90 

.91 

.92 

.93 

.94 

162 

.95 

.96 

.98 

.99 

1.00 

164 

.90 

.91 

.92 

.93 

.94 

164 

.95 

.96 

.98 

.99 

0.99 

166 

.91 

.92 

.93 

.93 

.94 

166 

.96 

.96 

.98 

.99 

.99 

168 

.92 

.93 

.93 

.94 

.94 

168 

.96 

.96 

.98 

.99 

.99 

170 

.93 

.94 

.94 

.95 

.95 

170 

.97 

.97 

.98 

.99 

.99 

172 

.94 

.95 

.95 

.96 

.96 

172 

.97 

.97 

.99 

.99 

.99 

174 

]  .95 

.96 

.96 

.96 

.97 

174 

.98 

.98 

.99 

.99 

.99 

176 

.97 

.97 

.97 

.97 

.98 

176 

.99 

.98 

.99 

.99 

.99 

178 

.98 

.98 

.99 

.99 

.99 

178 

.99 

.99 

.99   .99 

.99 

322 


Table  14.    Sumner  Intersection 


Smaller  Bearing 

Larger 
Bearing 

82° 

84° 

86° 

88° 

90° 

112° 

1.98 

2.12 

2.28 

2.46 

2.67 

114 

1.87 

1.99 

2.12 

2.28 

2.46 

116 

1.77 

1.88 

2.00 

2.13 

2.28 

118 

1.68 

1.78 

1.88 

2.00 

2.13 

120 

1.61 

1.69 

1.78 

1.89 

2.00 

122 

1.54 

1.62 

1.70 

1.79 

1.89 

124 

1.48 

1.55 

1.62 

1.70 

1.79 

126 

1.43 

1.48 

1.55 

1.62 

1.70 

128 

1.38 

1.43 

1.49 

1.55 

1.62 

130 

1.33 

1.38 

1.44 

1.49 

1.56 

132 

1.29 

1.34 

1.39 

1.44 

1.49 

134 

1.26 

1.30 

1.34 

1.39 

1.44 

136 

1.22 

1.26 

1.30 

1.34 

1.39 

138 

1.19 

1.23 

1.27 

1.30 

1.35 

140 

1.17 

1.20 

1.23 

1.27 

1.31 

142 

1.14 

1.17 

1.20 

1.24 

1.27 

144 

1.12 

1.15 

1.18 

1.21 

1.24 

146 

1.10 

1.13 

1.15 

1.18 

1.21 

148 

1.08 

1.11 

1.13 

1.15 

1.18 

150 

1.07 

1.09 

1.11 

1.13 

1.15 

152 

1.05 

1.07 

1.09 

1.11 

1.13 

154 

1.04 

1.06 

1.08 

1.09 

1.11 

156 

1.03 

1.05 

1.06 

1.08 

1.09 

158 

1.02 

1.03 

1.05 

1.06 

1.08 

160 

1.01 

1.02 

1.04 

1.05 

1.06 

162 

1.00 

1.01 

1.03 

1.03 

1.05 

164 

1.00 

1.00 

1.02 

1.02 

1.04 

166 

1.00 

1.00 

1.01 

1.02 

1.03 

168 

1.00 

1.00 

1.00 

1.01 

1.02 

170 

0.99 

1.00 

1.00 

1.00 

1.01 

172 

.99 

1.00 

1.00 

1.00 

1.01 

174 

.99 

0.99 

1.00 

1.00 

1.00 

176 

.99 

.99 

0.99 

1.00 

1.00 

178 

.99 

.99 

0.99 

0.99  1.00  1 

APPENDIX  V 

COMPASS  ADJUSTING 

In  Chapter  IV  we  have  assumed  that  the  ship's  compass 
will  be  properly  compensated  by  a  professional  compass 
adjuster  (p.  43),  and  that  the  navigator  will  thereafter  only 
need  to  check  the  adjuster's  table  of  small  remaining  devia- 
tions from  time  to  time  during  the  voyage.  This  occasional 
checking  is  accomplished  most  easily  by  observing  the  sun's 
azimuth  at  the  same  (or  very  nearly  the  same)  time  when  a 
sextant  altitude  is  measured  in  the  regular  work  of  navigat- 
ing the  ship  (cf.  p.  145). 

But  it  may  happen,  expecially  in  the  Navy,  that  the  navi- 
gator will  be  his  own  compass  adjuster :  he  may  be  re- 
quired to  swing  ship  (p.  43),  and  construct  a  complete  table 
of  deviations  himself.  To  do  this  he  will  probably  compare 
the  sun's  compass  bearing  with  its  true  azimuth  after  swing- 
ing the  ship's  head  successively  on  a  number  of  different 
courses.  Each  time  he  observes  the  sun's  bearing  with  a 
pelorus  (p.  44)  or  other  similar  instrument,  he  will  record 
the  time  by  his  watch,  which  should  as  usual  be  set  to  the 
ship's  apparent  time  (p.  94).  But  no  sextant  observations 
of  any  kind  will  be  needed ;  nor  will  the  sun's  altitude  ordi- 
narily be  calculated.  For  this  reason  it  is  impossible  to  ob- 
tain the  sun's  true  azimuth  from  our  Table  11  'p.  284)  which 
requires  a  knowledge  of  the  altitude,  and  which  is  merely 
intended  for  checking  the  compass  error  by  an  observation 
made  nearly  simultaneously  with  a  sextant  observation,  as 
just  explained. 

For  the  purposes  of  the  compass  adjuster,  the  sun's  true 
azimuth  is  most  conveniently  taken  from  Publication  71, 

323 


324  NAVIGATION 

U.  S.  Hydrographic  Office,  often  called  the  '^red"  azimuth 
table. ^  But  if  this  is  not  available  it  can  be  obtained  with 
almost  equal  ease,  and  without  interpolation,  from  the  Kel- 
vin Table  13  (p.  292),  the  use  of  which  is  in  this  case  greatly 
simplified  because  we  only  need  the  sun's  azimuth,  without  a 
*' computed  altitude"  (the  K3  of  p.  129),  and  because  the 
azimuth  itself  need  only  be  correct  to  within  a  degree. 
The  given  quantities  of  the  problem  are : 

1.  The  sun's  declination,  to  be  taken  to  the  nearest  degree  only, 

and  without  regard  to  its  +  or  —  sign ; 

2.  The  ship's  known  latitude,  or  D.  R.  latitude,  always  taken  to 

the  nearest  degree  only,  and  without  regard  to  sign,  except  when 
choosing  formulas ; 

3.  The  ship's  apparent  time,  taken  from  the  navigator's  watch; 

counted  for  the  present  purpose  in  civil  reckoning,  a.m.  or 
P.M.  (pp.  75,  78) ;  and  hereafter  called  "the  time." 

We  proceed  as  follows  :  ^ 

Operation  1.     Enter  Table  13  with: 

Arg.  ai  =  declination, 

Arg.  61  =  the  time,  if  it  is  earlier  in  the  morning  than  6  a.m.,  or 

earlier  in  the  afternoon  than  6  p.m.  ; 
Arg.  61  =  the  time  subtracted  from  12'',  if  later  than  6,  a.m.  or  p.m., 
and   before   use   61   must   be   turned  into  degrees  with 

Table   9    (p.    249).     It    need    be    correct    to     the 

nearest   degree   only ;    and   it   will    always   be   less 

than  90°. 

Then  take  from  Table  13  the  tabular  angle  Ki,  also  correct 
to  the  nearest  degree  only. 

Operation  2.     Enter  Table  13  a  second  time  with :     • 

Arg.  02  =  the  Ki  obtained  in  Operation  1. 

Then,  under  this  a2,  run  down  the  K-column  until  you  find 
the  K2  which  comes  nearest  to  the  declination ;  and  from 
the  left-hand  argument  column  take  the  62  which  is  in  the 

1  In  using  this  very  extended  table,  the  young  navigator  will  note 
that  the  words  "declination  —  same  name  as  —  latitude"  signify 
that  declination  and  latitude  have  the  same  sign,  both  +  or  both  — . 

2  This  is  a  modification  of  the  proceeding  of  p.  127. 


COMPASS  ADJUSTING  325 

same  horizontal  line  with  the  declination  K2  just  found  in 
the  X-column. 

Operation  3.  Add  62  to  the  given  latitude,  and  call  it 
the  sum.  Also  take  the  difference,^  between  62  and  the  lati- 
tude, subtracting  the  smaller  from  the  larger.  Then  enter 
Table  13  a  third  time  with : 

Arg.  as  =  Ki,  again  as  obtained  in  Operation  1. 
(5^  Arg.  63  =  90°  —  above  sum,  if  latitude  and  declination  are  of 

opposite  signs,  one  +  and  one  — . 
(6')  Arg.  63  =  above  sum  —  90°,  if  the  time  was  later  than  6  p.m. 

in  the  afternoon,  or  earlier  than  6  a.m.  in  the 

morning. 
(?')  Arg.  63  =  90°  —  above  difference,  in  all  other  cases. 

Then  with  the  arguments  as  and  63,  take  from  Table  13 
the  tabular  Q3,  the  sun's  true  azimuth,  to  the  nearest  degree. 
If  the  latitude  is  + ,  this  azimuth  Q3  is  to  be  counted  from  the 
north  point  of  the  horizon  if  we  used  formula  (6')  just  given ; 
or  if,  in  using  formula  (7'),  &2  was  greater  than  the  latitude ; 
otherwise  Qz  is  to  be  counted  from  the  south  point  of  the 
horizon.  (If  the  latitude  is  — ,  interchange  the  north  and 
south  points  of  the  horizon  in  these  directions. 2)  And  in 
all  latitudes,  the  azimuth  will  of  course  be  counted  toward 
the  east  or  west,  according  as  the  time  was  a.m.  or  p.m. 

The  foregoing  will  enable  the  navigator  to  obtain  the 
sun's  true  azimuth  from  Table  13,  either  for  compass  adjust- 
ing purposes,  or  in  case  he  should  ever  wish  to  know  the 
azimuth  when  no  altitude  has  been  observed.  The  follow- 
ing are  examples  :     Given : 

1.  Dec.  =  +  8°  ;  D.  R.  lat.  =  +  38°  ;  ship's  apparent  time 
=  4''  10"*,  P.M.;  ship's  head  by  compass  =  165°;  observed 
bearing  of  sun  =  240°. 5. 

*  The  sum  and  difference  are  not  both  needed ;  usually  only  one 
of  the  two  will  be  written  down. 

2  It  will  not  usually  be  necessary  to  consider  these  directions 
about  Q3,  because  the  navigator  will  generally  know  whether  the 
sun  bore  N.  or  S.  of  the  E.  or  W.  point  of  the  horizon  at  the  time  of 
observation. 


326  NAVIGATION 

Operation  1  gives  ai  =  8°;     6i  =  4''  10"^  =  621°   (p.  249); 

Ki  =  61°(p.  295); 
Operation  2  gives  az  =  61°;  X2  =  8°;  62  =  17°  (p.  308) ; 
Operation  3  gives  sum  =  55°;  difference  =  21°;  as  =  61°; 
63  =  69°;  Q3  =  79°;  sun's  azimuth  =;S  79°  W  =  259°. 
The  red  tables,  p.  88,  give  N  101°  W.  =  259°.  Then  by 
formula  (2),  p.  45,  we  have:  E  =  T  -  C  =  259°  -  240°.5 
=  +  18°. 5  =  compass  error.  And  if  we  take  the  variation 
to  be  +  10°,  as  on  p.  48,  we  have  by  formula  (1),  p.  45, 
D  =  E  -  V  =  18°.5  -  10°  =  +  8°.5  =  the  deviation  when 
the  bearing  of  the  ship's  head  by  compass  was  165°.  This 
deviation  is  the  same  as  is  given  in  the  table  on  p.  48. 

2.  Dec.  =   -  8°;   D.  R.  lat.  =  +38°;   time  =  7''  50"*,  a.m.; 

ship's  head  by  compass  =  75°;  compass  bearing  of  sun  =  114°; 
ai  =  8°;  61  =  12''  -  7'«  50^"  =  4»  10"»  =  62i°;  Ki  =  61°; 
a2  =  61°;  X2  =  8°;  62  =  17°; 

swm  =55°;  diff.  =21° ;  03  =61°;  63=35°;  Q3=S66°E  =  114°. 
The  red  tables  also  give  114°  for  the  sun's  azimuth,  afford- 
ing an  excellent  check  on  the  work.  Now  the  compass  error 
E  =  T  -  C  =  114°  -  114°  =  0°.  With  V  =  +  10°, 
D  =  J5;  -  F  =  0°  -  10°  =  -  10°.  The  table  on  p.  48  gives 
D  =  -  9°.7. 

3.  Dec.  =  +15°;    D.  R.  lat.  =  +38°;    time  =  5*  40'",  a.m.; 

ship's  head  by  compass  =  225°;  compass  bearing  of  sun  =39°; 
a,  =  15°;  bi  =  5^^  40-  =  85°;  Ki  =  74°; 
02  =  74°;  K2  =  15°;  b^  =  70°; 

st^m  =  108°;  di^.  =32°;  03  =74°;  63  =  18°;  Q3  =N  75°  E  =75°. 
The  red  tables  also  give  75°  for  the  sun's  azimuth.  And 
the  compass  error  E  =  T  -  C  =  75°  -  39°  =  36°.  With 
F  =  +  10°,  D  =  E  -  F  =  36°  -  10°  =  +  26°.  The  table 
on  p.  48  gives  D  =  +  25°.6. 

In  this  way  the  entire  deviation  table  of  p.  48  might  have 
been  obtained  from  observations,  and  the  Second  Deviation 
Table  (p.  49)  subsequently  computed. 

In  connection  with  these  two  deviation  tables,  it  may  be 
of  interest  to  supplement  p.  49  by  emphasizing  once  more  that 
both  tables  are  needed  in  correct  navigation.  The  second 
table  is  necessary  for  changing  a  true  course  into  a  compass 
course  for  the  helmsman  (see  p.  143  for  an  example)  :  and  the 
first  table   (in   coastwise  navigation)   for  correcting  a  re- 


COMPASS   ADJUSTING  327 

versed  bearing  (p.  55),  or  fixing  a  ship's  position  by  cross 
bearings  (p.  56).  Only  if  the  compass  has  been  very  well 
compensated  or  adjusted  is  it  permissible  to  navigate  with 
one  table  only.  With  a  compass  thus  compensated  the  out- 
standing deviations  would  be  so  small  that  the  two  tables 
would  be  practically  interchangeable.  Were  it  possible  to 
effect  a  perfect  compensation,  the  two  tables  would  be 
identical,  and  all  the  deviations  of  both  would  be  0°. 

Having  now  explained  the  method  of  determining  devia- 
tions without  measuring  or  calculating  the  sun's  altitude, 
we  shall  next  consider  in  a  practical  way  the  principal  prob- 
lem of  compass  adjusting,  or  the  placing  of  magnetic 
and  other  correctors  in  position,  so  as  to  minimize  the  de- 
viation on  all  courses.  We  shall  begin  with  certain  defini- 
tions. ' 

1.  Semicircular  deviation  is  that  part  of  the  total  devia- 
tion which  is  corrected  by  two  permanent  magnets  (or  bun- 
dles of  thin  magnets)  placed  in  the  lower  part  of  the  binnacle. 
One  of  these  permanent  magnets  is  always  placed  in  a  fore- 
and-aft  position,  the  other  in  a  thwartship  position.  Both 
may  be  raised  and  lowered,  so  as  to  change  their  distances 
from  the  compass  card.  The  north  (or  north-seeking)  ends 
of  all  permanent  magnets  are  always  painted  red. 

2.  Quadrantal  deviation  is  that  part  of  the  total  devia- 
tion which  is  corrected  with  two  hollow  iron  spheres  or  other 
pieces  of  iron  placed  on  each  side  of  the  compass  bowl  in  an 
athwartship  direction.  They  are  adjustable  in  position,  so 
that  their  distances  from  the  compass  card  can  be  varied. 

3.  The  heeling  error  is  an  additional  deviation  caused  by 
the  ship's  rolling,  and  is  corrected  with  an  additional  perma- 
nent magnet  placed  in  a  vertical  position  directly  under  the 
center  of  the  compass  bowl. 

4.  The  following  procedure  may  be  used  on  a  compass 
entirely  uncompensated,  or  on  a  compass  already  approxi- 
mately compensated,  either  by  actual  observations,  or  by 
the  placing  of  magnets  in  approximate  positions  suggested 


328  NAVIGATION 

by  experience.  The  method  is  specially  designed  to  avoid 
the  necessity  of  steering  directly  by  the  sun,^  by  ranges  of 
known  bearing,  or  by  means  of  a  ''  Napier  diagram,"  in  the 
course  of  the  adjustment. 

5.  With  the  ship  on  an  even  keel  and  all  permanent  mag- 
nets being  removed,  begin  by  moving  the  vertical  heeling 
magnet  from  top  to  bottom  of  its  travel.  This  should  not 
affect  the  compass  card  at  all.  If  it  does,  the  compass  bowl 
is  itself  not  properly  centered  in  the  binnacle,  and  its  position 
there  must  be  adjusted  by  the  proper  adjusting  screws. 

6.  After  the  preliminary  centering  under  5,  remove  the 
heeling  magnet  to  a  distance,  and  place  the  two  iron  spheres 
in  an  approximately  proper  position,  suggested  by  experi- 
ence; or,  if  lacking  experience,  place  them  in  the  middle 
positions  permitted  by  their  respective  ranges  of  adjust- 
ment. 

7.  Next  you  must  learn  how  to  head  your  ship  on  any  de- 
sired magnetic  course,  say  M.  To  do  this,  let  G  represent 
any  convenient  auxiliary  number  of  degrees.  In  a  steel  ship, 
with  compass  entirely  uncompensated,  we  might  put  G  =  15°. 
In  a  wooden  ship,  or  for  a  compass  already  approximately 
compensated,  we  might  take  G  =  10°,  or  even  less.  In 
general,  G  should  be  about  half  as  large  as  the  largest  re- 
maining deviations  the  compass  is  expected  to  have. 

Now  steady  the  ship  on  the  compass  course  M  —  G,  and 
keep  her  steady  on  that  course  by  heading  for  some  object 
ashore,  or  by  careful  use  of  the  compass.  While  running 
slowly  on  that  course,  observe  the  sun's  compass  bearing  and 
note  the  ship's  apparent  time  by  your  watch.  The  watch 
should  be  set  in  advance  to  ship's  apparent  time  (see  p.  94). 

Then,  with  the  red  azimuth  tables,  or  the  Kelvin  table, 
ascertain  the  true  bearing  of  the  sun,  which  we  will  call  T, 
and  calculate  the  compass  error  E  =  T  —  {M  —  G).  The 
variation,  F,  being  taken  from  the  chart,  you  will  have  the 

1  •'  Maneuver  the  ship  with  the  helm  until  the  sun  comes  on  the 
sight  vanes  (of  the  pelorus)."     Bowditeh,  p.  51,  1916  edition. 


COMPASS  ADJUSTING  329 

deviation  D  =  T  —  {M  —  G)  -  V.  Call  this  deviation  di 
(it  corresponds  to  the  compass  course  M  —  G). 

Now  steady  the  ship  on  a  new  compass  course  M  -\-  Gj 
and  determine  by  observation  in  exactly  the  same  way  a 
new  deviation,  which  call  ^2- 

You  will  then  have  : 


For  ship's  head  by  compass 

the  deviation 

M  -G, 

du 

M  +  G, 

(h. 

Then  the  deviation  for  the  magnetic  course  M,  which  we 
desire  to  find,  and  which  we  will  call  d^,  will  be : 

G(d2  +  d,) 
""      2G  +  d2-  di' 

And  the  required  compass  course,  Cm,  corresponding  to 
the  given  magnetic  course  M,  will  be : 

Cm  =  M  -  dM. 

The  value  of  dM  may  be  taken  from  the  accompanying  little 
Table  in  all  cases  that  are  likely  to  arise  in  actual  work. 
Should  a  number  ever  be  required  from  a  blank  place  in  the 
Table,  the  compass  probably  has  unusual  deviations,  and  a 
preliminary  partial  compensation  should  be  attempted  by 
means  of  known  ranges  taken  from  a  chart. 

8.  Go  through  the  work  under  7  for  the  magnetic  course 
M  =  0°  (or  due  north).  If  you  take  G  =  15°,  this  will 
necessitate  determining  by  observation  the  deviations  di 
and  ^2  for  the  compass  courses  0°  —  15°  =  345°,  and 
0°  +  15°  =  15°  (see  example,  p.  333). 

You  will  then  calculate  do  and  Co,  the  deviation  and 
compass  course  corresponding  to  the  magnetic  course  0°, 
using  the  above  formula  for  dM,  which  in  this  case  is  c^ ;  or 
you  will  take  do  directly  from  the  Table. 

9.  Steady  your  ship  on  this  compass  course  Co  (or  magnetic 
course  M  =  0°),  and  keep  her  quite  steady  by  heading  for 
a  visible  fixed  point  like  a  light-house,  or  by  using  tem- 


330 


NAVIGATION 


Values  of  dm,  the  Deviation  for  the  Magnetic  Course  M 


G  =  16° 


da,  THE 

Deviation  for  the  Compass  Course  M  +  G 

-30° 

-26° 

-20° 

-19° 

-16° 

-10° 

-6° 

0° 

+5° 

+10° 

+16° 

+20° 

+26° 

+30° 

O    -30° 

-30° 

-24° 

-15° 

-12° 

-10° 

-  8° 

-   6° 

-  4° 

-  3° 

-  2° 

-   1° 

0° 

1     -25 
■^    -20 

-33 

-25 

-19 

-15 

-12 

-  9 

-  7 

-  5 

-  4 

-  2 

-   1 

0 

+  1 

-27 

-20 

-15 

-11 

-  8 

-  6 

-  4 

-  2 

-   1 

0 

+  1 

+  2 

I    -15 

1  -10 

-30 

-21 

-15 

-11 

-  8 

-   5 

-  3 

-   1 

0 

+  1 

+  2 

+  3 

-22 

-15 

-10 

-   6 

-   4 

-  2 

0 

+  1 

+  2 

+  4 

+  4 

^    -   5 

-25 

-15 

-  9 

-  5 

-  2 

0 

+  2 

+  3 

+  4 

+  5 

+  6 

a      0 

-30 

-15 

-  8 

-  3 

0 

+  2 

+  4 

+  5 

+  6 

+  7 

+  8 

S    +  5 

-15 

-  5 

0 

+  3 

+  5 

+  6 

+  8 

+  8 

+  9 

+10 

S  +10 

+30 

-15 

0 

+  5 

+  8 

+  9 

+10 

+11 

+  11 

+  12 

+  12 

i  +15 

+  15 

+15 

+15 

0 

+15 

+15 

+  15 

+  15 

+  15 

+  15 

+  15 

+  15 

+  15 

'>    +20 

+  8 

+  5 

0 

-15 

+30 

+25 

+22 

+21 

+20 

+19 

+19 

Q    +25 

+  3 

0 

-   5 

-15 

+35 

+30 

+27 

+25 

+23 

^-   +30 

0 

-  3 

-  8 

-15 

-30 

+33 

+30 

G  =  10° 


d2,  THE  Deviation  for  the  Compass  Course  M  -\-  G 

-30° 

-26° 

-20° 

-16° 

-10° 

-6° 

0° 

+5° 

+10° 

+16° 

+20° 

+26° 

+30° 

^"    -30° 

-30° 

-22° 

-17° 

-13° 

-10° 

-  8° 

-  6° 

-   5° 

-  3° 

-  2° 

-    1° 

-   1° 

0° 

'     -25 
■^    -20 

-37 

-25 

-18 

-13 

-10 

-  8 

-  6 

-  4 

-  3 

-   2 

-   1 

0 

+  1 

-30 

-20 

-14 

-10 

-   7 

-  5 

-  3 

-  2 

-   1 

0 

+  1 

+  1 

e  -15 
t  -10 

-23 

-15 

-10 

-  7 

-  4 

-  3 

-   1 

0 

+  1 

+  2 

+  2 

-30 

-17 

-10 

-  6 

-  3 

-    1 

0 

+   1 

+  2 

+  3 

+  3 

^    -   5 

-20 

-10 

-   5 

-  2 

0 

+  1 

+  2 

+  3 

+  4 

+  4 

a      0 

+30 

-30 

-10 

-  3 

0 

+  2 

+  3 

+  4 

+  5 

+  6 

+  6 

6  +  5 

+  17 

+20 

+30 

-10 

0 

+  3 

+  5 

+  6 

+  7 

+  7 

+  8 

+  8 

g    +10 

+  10 

+10 

+  10 

+10 

0 

+  10 

+  10 

+  10 

+  10 

+10 

+  10 

+  10 

+  10 

fl    +15 

+  6 

+  5 

+  3 

0 

-10 

+30 

+20 

+  17 

+  15 

+  14 

+  13 

+  13 

\    +20 

+  3 

+  2 

0 

-  3 

-10 

-30 

+30 

+23 

+20 

+  18 

+17 

Q    +25 

+  1 

0 

-  2 

-   5 

-10 

-20 

+30 

+25 

+22 

^-   +30 

0 

-   1 

-   3 

-  6 

-10 

-17 

-30 

+30 

G  =  I 

i° 

di,  THE  Deviation  for  the  Compass  Course  M  +  G 

-30° 

-26° 

-20° 

-16° 

T-10° 

-6° 

0° 

+6° 

+10° 

+16° 

+20° 

+26° 

0° 

+30° 

O    -30° 

-30° 

-18° 

-12° 

-   9° 

-  7° 

-  5° 

-   4° 

-  3° 

-  2° 

-   1° 

-    1° 

0° 

1     -25 

-25 

-15 

-10 

-  7 

-  5 

-   4 

-  2 

-  2 

-   1 

0 

0 

0 

^    -20 

-20 

-12 

-  8 

-  5 

-  3 

-  2 

-   1 

-   1 

0 

0 

+  1 

S    -15 

-35 

-15 

-  8 

-  5 

-  3 

-  2 

-   1 

0 

+  1 

+  1 

+  1 

3    -10 

+20 

+35 

-25 

-10 

-   5 

-  2 

-   1 

0 

+  1 

+  1 

+  2 

+  2 

O    -    5 

+  12 

-15 

+25 

-15 

-  5 

-  2 

0 

+  1 

+  2 

+  2 

+  2 

+  3 

a      0 

+  8 

+  8 

+  10 

+  15 

-   5 

0 

+  2 

+  2 

+  3 

+  3 

+  4 

+  4 

6  +  5 

+  5 

+  5 

+  5 

+  5 

+  5 

0 

+  5 

+  5 

+  5 

+  5 

+  5 

+  5 

+  5 

g    +10 

+  3 

+  3 

+  2 

+  2 

0 

-   5 

+  15 

+  10 

+  8 

+  8 

+  7 

+  7 

i  +15 

+  2 

+  2 

+   1 

0 

-  2 

-   5 

-15 

-25 

+  15 

+  12 

+  10 

+  9 

\    +20 

+  1 

+  1 

0 

-   1 

-  2 

-   5 

-10 

-25 

+20 

+  15 

+  12 

Q    +25 

+  1 

0 

-    1 

-  2 

-  3 

-  5 

-  8 

-15 

-35 

+25 

+  18 

^-   +^ 

0 

-   1 

-    1 

-   2 

-  3 

-  5 

-  8 

-12 

-20 

+30 

COMPASS  ADJUSTING  331 

porarily  an  auxiliary  compass.  But  this  auxiliary  compass 
must  not  be  near  enough  to  the  magnets  to  be  influenced 
by  them. 

10.  Move  the  thwartship  permanent  correcting  magnet 
toward  or  from  the  compass  bowl,  until  the  lubber  line  (p. 
42)  is  on  the  correct  magnetic  course  0°.  If  you  are  working 
with  a  compass  as  yet  entirely  uncompensated,  for  which  the 
permanent  magnets  have  not  even  been  placed  in  the  bin- 
nacle, the  thwartship  one  should  be  located  with  its  red  end 
to  starboard,  if  the  do  found  under  8  was  plus,  or  easterly 
deviation  ;  and  with  its  red  end  to  port,  if  that  do  was  minus, 
or  westerly  deviation. 

11.  Go  through  the  work  under  7  again  for  the  magnetic 
course  M  —  90°  (or  due  east).  This  will  necessitate  deter- 
mining by  observation  the  deviations  for  the  compass  courses 
75°  and  105°,  if  you  are  working  with  G  =  15°.  And  you 
will  calculate  d^o  and  C90,  the  deviation  and  compass  course 
for  the  magnetic  course  90°. 

12.  Now  steady  the  ship  on  the  compass  course  C90,  and 
place  the  fore-and-aft  compensating  permanent  magnet  with 
its  red  end  forward,  if  the  d^o  found  under  11  was  plus,  and 
with  its  red  end  aft,  if  dgo  was  minus.  Adjust  the  magnet 
so  as  to  make  the  compass  read  90°.  Your  semicircular 
deviation  is  now  corrected. 

13.  Go  through  the  work  under  7  for  the  magnetic  course 
M  =  45°  (or  north-east,  magnetic).  This  will  necessitate 
observing  the  sun  on  the  compass  courses  30°  and  60° ;  and 
will  give  you  d^b  and  C45,  the  deviation  and  compass  course 
corresponding  to  magnetic  course  45°. 

14.  Steady  your  ship  on  the  compass  course  C45,  and  move 
the  two  spheres  in  and  out  until  the  lubber  line  is  on  45°, 
leaving  the  two  spheres  finally  so  placed  that  they  are 
equally  distant  from  the  compass  bowl.  Your  quadrantal 
deviation  is  now  corrected. 

15.  To  compensate  for  heeling  error,  head  the  ship  approx- 
imately north  or  south,  and  keep  her  accurately  on  that 


332  NAVIGATION 

course  by  heading  slowly  for  an  object  ashore.     Now  heel 
the  vessel  about  10°,  by  any  convenient  method. 

If  the  north-seeking  end  of  the  compass  card  is  thereby 
deviated  toward  the  high  side  of  the  ship,  place  the  heeling 
corrector  with  red  end  up  in  such  a  position  as  will  bring  the 
compass  card  back  where  it  was  before  ship  was  heeled. 
If  the  compass  card  was  deviated  toward  the  low  side  of  the 
ship,  place  the  heeling  corrector  with  the  red  end  down. 

16.  The  ''Flinders  bar"  is  a  vertical  bar  of  soft  iron  (or  a 
combination  of  several  bars)  sometimes  placed  directly  for- 
ward or  aft  of  the  compass.  It  will  correct  a  certain  part 
of  the  semicircular  deviation  not  fully  removed  by  the  per- 
manent magnets  adjusted  under  10  and  12.  Usually  a 
Flinders  bar  is  best  located  by  placing  it  in  a  position  sug- 
gested by  experience;  but  many  compasses  are  adjusted 
without  such  a  bar,  and  v/hen  there  is  none,  the  magnets 
usually  need  readjustment  whenever  the  ship  changes  her 
latitude  very  considerably. 

17.  After  completing  the  adjustment,  it  is  well  to  swing 
ship  on  eight  equidistant  courses,  and  check  the  deviation 
table  by  new  observations. 

18.  After  a  compass  has  once  been  adjusted,  necessary 
minor  changes  of  the  magnets  and  spheres  can  be  most  con- 
veniently made  as  follows.  Head  the  ship  north,  and  steady 
her  with  an  auxiliary  compass,  or  by  means  of  a  conspicuous 
object  ashore.  Then  move  the  athwart  ship  magnet  up  one 
inch,  and  note  by  the  compass  bearing  of  the  sun  how  much 
the  compass  has  changed,  and  in  which  direction.  The  same 
thing  can  be  done  with  the  fore-and-aft  magnets  by  heading 
the  ship  east ;  and  with  the  spheres  by  heading  northeast. 
Having  thus  ascertained  how  much  the  compass  is  changed 
by  a  one-inch  motion  of  each  corrector,  it  is  easy  to  calculate 
how  much  they  should  each  be  moved  to  compensate  for  any 
outstanding  small  deviations  on  the  north,  east,  and  north- 
east magnetic  courses.  Corrections  can  thus  be  made  at  any 
time  during  a  voyage,  if  the  deviations  become  unduly  large. 


COMPASS  ADJUSTING  333 

When  the  magnets  are  not  movable,  but  consist  of  fixed 
bundles  of  thin  wire  magnets,  all  adjustments  throughout 
are  made  by  increasing  or  diminishing  the  number  of  wires, 
instead  of  moving  the  magnets  toward  the  compass  bowl 
or  away  from  it. 

Notes 

Note  to  8.  You  can  equally  well  head  the  ship  south  in- 
stead of  north,  and  go  through  the  work  for  M  =  180°,  in- 
stead of  M  =  0°. 

Note  to  10.  If  you  head  south,  according  to  the  Note  to  8, 
the  red  end  of  the  thwart  ship  magnet  must  lie  reversed. 

Note  to  11.  This  work  may  be  done  before  that  under  8, 
if  desired. 

Note  to  12.  You  may  head  the  ship  west,  if  you  wish, 
instead  of  east,  and  work  for  M  =  270°,  instead  of  90°.  The 
magnet  must  then  be  placed  with  red  end  aft,  to  correct  plus 
deviation. 

Note  to  14.  This  may  equally  well  be  done  for  M  =  135°, 
225°,  or  315°. 

Note  to  18.     The  above  notes  to  12  and  14  also  apply  to  18. 

General  Note.  Whenever  an  adjustment  can  be  made  on 
two  opposite  courses,  as  indicated  in  the  above  Notes,  accu- 
racy will  be  increased  by  adjusting  on  both  courses,  and 
leaving  the  correctors  finally  in  the  average  of  the  two  posi- 
tions found. 

EXAMPLE 

Consider  the  compass  for  which  the  two  deviation  tables  (pp.  48, 
49)  hold  good ;  and  we  shall  suppose  it  to  have  been  a  totally  un- 
compensated compass. 

Under  8  and  7,  putting  M  =  0°,  G  =  15"*,  we  have : 
for  compass  course  M  —  G  =  345°,  di  =  —  16°.0  (table,  p.  48), 
for  compass  course  M  +  G  =  15**,    d2  =  —  14°. 9  (table,  p.  48). 

Then,  dM=do=  Mdi±d^  =  15X(-3Q.9)    ^  _463^  ^  _^^.^^ 
2G-{-d2-di      30-14.9  +  16.0  31.1 

This  —  14°.9  is  in  exact  agreement  with  the  do  given  in  the  second 
deviation  table   (p.  49),  for  the  magnetic  course  M  =  0°.     The 


334  NAVIGATION 

agreement  would  not  always  be  as  perfect.     The  —  14°.9  must  now 
be  corrected  with  the  thwartship  magnet  as  directed  under  10. 

Next,  under  11,  for  M  =  90°,  we  have : 
for  compass  course  M  —  G  =    75°,  di  =   —  9°. 7  (table,  p.  48), 
for  compass  course  M  +  G  =  105°,  6,2=-  9°.0  (table,  p.  48). 

Then,  dM  -  d.o  =    Gid,±d^  ^  15X(-18.7)  ^  _  2805  ^  .^o  ^^ 
2G-\-d2-di      30-9.0+9.7  30.7 

The  —  9°.l  agrees  closely  with  —  9°.0,  given  in  the  second  devia- 
tion table  (p.  49)  for  M  =  90°.  It  must  be  corrected  as  directed 
under  12.     This  completes  the  ordinary  semicircular  compensation. 

Coming  now  to  13,  with  M  =  45°,  we  must  observe  the  sun  on  the 
compass  courses  30°  and  60°.  But  the  semicircular  correction  being 
now  complete,  the  observed  deviations  will  no  longer  agree  with 
those  given  in  the  table,  which  are  supposed  to  have  been  observed 
with  a  compass  entirely  uncompensated. 

Let  us  suppose  the  observations  gave  the  following  results : 

for  compass  course  M  —  G  =  30°,  rfi  =  +  6°.9, 
for  compass  course  M  -\-  G  =  60°,  d2  =  +  6°.0. 

Then,  dM=d.,=    ^^^-+^-^    =      ^^X^^'^      =+l^  =  +Qoe 
2G-{-d2-di      30+6.0-6.9  29.1 

This  6°.6  must  now  be  corrected  as  directed  under  14,  completing 
the  quadrantal  compensation. 


APPENDIX  2 

EX-MERIDIAN   AND  MISCELLANEOUS  EXAMPLES 

Ex -MERIDIAN  observations  (p.  99)  are  completely  and  ac- 
curately calculated  with  the  Kelvin  Table  13,  working  out  a 
Sumner  line  (see  p.  148  for  an  example).  But  if  a  rapid  cal- 
culation of  the  ship's  latitude  only  is  desired,  we  may  either 
use  special  tables  (p  99,  footnote),  or,  if  these  are  not  avail- 
able, we  may  apply  the  Kelvin  Table  with  but  little  additional 
labor  and  almost  equal  accuracy.  We  may  still  use  the 
simplified  method  already  explained  in  Appendix  1  (p.  324) ; 
except  that  Q3  will  not  now  be  required,  and  K2  as  well  as  K3 
must  be  taken  from  the  Table  exact  to  the  nearest  minute 
(see  Ex.  1)  This  having  been  done,  the  ship's  latitude, 
at  the  moment  of  observation,  may  be  quickly  calculated  from 
the  ex-meridian  altitude  by  first  choosing  from  p.  89  the 
formula  which  would  be  appropriate  for  a  noon-sight,  and 
then  applying  to  the  D.  R.  latitude  {taken  to  the  nearest  degree 
only)  the  two  following  corrections : 

the  "altitude  correction"         =  corrected  observed  altitude  —  K3; 
the  "declination  correction"  =  sun's  declination  —  K2. 

These  corrections  are  to  be  added  or  subtracted,  according 
as  the  formula  chosen  from  p.  89  had  a  +  or  —  sign  for  the 
altitude  and  declination  respectively.  This  is  the  only  use 
here  made  of  the  formula. 

Young  naval  officers  having  commands  should  give  special 
attention  to  the  foregoing,  because  they  may  be  required  to 
signal  their  latitude  to  the  flagship  promptly  at  noon,  before 
they  have  had  time  to  calculate  a  noon-sight.  In  such  cases 
an  ex-meridian  taken  at  about  1 P  30"*,  shipls  apparent  time, 

335 


336  NAVIGATION 

and  the  resulting  latitude  carried  forward  to  noon  with  the 
traverse  table,  will  furnish  an  excellent  value  for  the  noon 
latitude  to  be  signaled.  The  whole  calculation,  including 
the  carrying  forward  to  noon,  can  be  completed  in  a  few 
minutes,  and  the  signal  flags  bent  on,  ready  to  be  run  up  at 
noon  precisely.  The  navigator  will  then  be  free  to  observe 
a  noon-sight  as  a  check. 

As  the  noon  longitude  is  always  signaled  as  well  as  the 
latitude,  a  time-sight  should  be  observed  (if  weather  permits) 
in  the  early  morning.  This  time-sight  should  be  calculated 
as  a  Sumner  long  before  noon ;  and  the  resulting  Sumner 
line  should  be  carried  forward  to  noon  by  D.  R.  methods 
(p.  137),  estimating  in  advance  the  probable  speed  of  the  ship 
and  her  course  to  noon.  An  ex-meridian  observation  made 
at  about  IP  30"*  (and  also  carried  forward)  having  furnished 
the  noon  latitude,  the  complete  noon  position  of  the  ship 
will  be  finally  fixed  at  that  point  of  the  moved  Sumner  line 
which  cuts  the  ship's  noon  parallel  of  latitude  (see  Ex.  4). 
But  when  the  navigator  is  not  hurried  by  the  necessity  of 
signaling  the  ship's  position  at  noon,  it  is  better  to  work 
out  a  Sumner  line  from  the  morning  time-sight,  and  also  from 
a  sight  taken  near  noon  (or  at  noon),  and  then  determine  the 
intersection  point  of  the  two  Sumner  lines  in  the  regular  way. 

Ex.  1.  Observed  altitude,  26°  55' ;  index,  +3';  height  of  eye, 
15  feet ;  watch  time  of  observation,  11*  42"*  0*  a.m.  ;  D.  R.  latitude, 
to  the  nearest  degree,  39° ;  D.  R.  longitude,  73°  58' ;  C.  -  TF., 
4/1  5im  42s ;  chron.  slow,  4*;  equation,  +  3"»  22«;  declination, 
—  23°  24' ;  find  the  latitude  by  the  ex-meridian  method.  (This  is 
the  example  worked  as  a  Sumner  on  pp.  148-149.) 

The  corrected  observed  altitude  comes  out  27°  8' ;  ship's 
apparent  time,  ll''  41"*  16«  a.m.;  ai  =  23°;  61  =  18"*  44» 
=  4°  41'  =  5°,  to  the  nearest  degree ;   i^i  =  4° ;  ^    as  =  4° ; 

^  The  value  4°  is  the  nearest  whole  degree  for  Ki,  since,  in  using 
Table  13,  we  notice  that  61  was  only  4°  4l',  and  therefore  not 
quite  5°.  But  our  result  would  be  almost  as  accurate  if  we  con- 
tinued the  calculation  with  Ki  =  5°  (see  also  Ex.  11). 


MISCELLANEOUS   EXAMPLES  337 

K2  =  22°  56'  (taken  out  to  the  nearest  minute) ;  62  =  23° ; 
sum  =  62°;  63  =  90°  -  sum  =  28°;  as  =  4° ;  K,=  27°  56' 
(taken  to  the  nearest  minute).  We  choose  formula  (4), 
p.  89,  or  lat.  =  90°  —  alt.  —  dec.  The  altitude  correction 
is  27°  8'  -  27°  56'  =  -  48',  which  must  be  subtracted,  be- 
cause alt.  is  —  in  the  formula.  The  declination  correction 
is  23°  24'  -  22°  56'  =  +  28',  which  must  also  be  subtracted, 
because  dec.  is  also  —  in  the  formula.  The  D.  R.  latitude 
being  39°,  the  final  latitude  will  be  39°  -  (-  48')  -  28'  = 
39°  20'.  On  p.  149  we  found  39°  19'  by  the  Sumner  calcu- 
lation. 

Ex.  2.  Corrected  observed  ex-meridian  altitude,  74°  26' ;  ship's 
apparent  time,  12^  24^"  p.m.  ;  declination,  +  3°  12';  D.  R.  latitude, 
+  17°  45',  or,  to  the  nearest  degree,  +  18°.  Find  the  latitude. 
Ans.  17°  39'. 

Ex.  3.^  Corrected  observed  ex-meridian  altitude,  72°  3' ;  ship's 
apparent  time,  ll''  46"*  a.m.  ;  declination,  +  20°  30' ;  D.  R.  latitude, 
+  3°  5' ;  find  the  latitude.     Ans.    2°  53'. 

Ex.  4.  At  sea,  at  9'^  42"^  28^^  a.m,  by  the  watch  (see  p.  146),  a 
time-sight  was  observed,  and  worked  as  a  Sumner.  It  gave  a  Sum- 
ner point  in  lat.  39°  50'  N.,  long.  73°  56'  W.,  bearing  of  Hne,  237°. 
The  ship  was  estimated  to  be  steaming  at  a  speed  of  15  knots  on  a 
true  course  of  182°.  At  IP  42'"  an  ex-meridian  (see  Ex.  1)  gave  the 
latitude  39°  20'.  Find  the  latitude  and  longitude  to  be  signalled 
at  noon. 

Ans.  Sumner  point  carried  forward  to  noon  is  then  in 
lat.  39°  16',  long.  73°  58' ;  bearing  of  line  unchanged  at  237°. 

1  If  the  observed  altitude  is  larger  than  45°,  it  is  well  to  be  spe- 
cially careful  in  taking  out  K3.  For  instance,  if  Ki  happened  to  be 
3^°,  a2  as  well  as  a^  would  also  be  3^°,  and  we  might  therefore  take 
K2  and  K3  from  the  column  headed  a  =  3°  or  the  column  headed 
a  =  4°.  In  the  case  of  sun  observations  the  choice  between  the 
two  columns  will  not  matter  for  K2,  but  for  K3  it  is  better  to  inter- 
polate between  the  values  given  in  the  two  adjoining  columns  in 
question  (see  Ex.  3). 

It  may  also  help  the  beginner  in  choosing  between  the  sum  and 
difference  formulas  of  p.  325  to  remember  that  the  proper  formula 
will  always  make  63  come  within  a  degree  or  two  of  the  observed 
altitude  in  the  case  of  ex-meridian  observations. 


338  NAVIGATION 

The  ex-meridian  carried  forward  to  noon  gives  the  ship's 
noon  latitude  as  39°  15'  (to  be  signaled).  So  the  latitude 
difference  at  noon  between  the  ship  and  the  Sumner  point 
is  1',  and  the  bearing  of  the  ship  from  the  Sumner  point  is 
237°.i  For  course  237°  and  lat.  diff.  1',  the  Traverse  Table 
gives  dep.  =  1'.7.  The  corresponding  long.  diff.  is  2'. 2; 
and  so  the  ship's  long,  at  noon  =  73°  58'  +  2'  =  74°  0'  (to 
be  signaled). 

Ex.  5.  At  sea,  Sept.  20,  1918,  a.m.,  with  D.  R.  lat.  45°  26'  N. ; 
D.  R.  long.  21°  40'  W. ;  at  7*  58'"  26%  a.m.  by  the  watch,  the  sun's 
measured  altitude  was  22°  7' ;  index,  +  3' ;  height  of  eye,  26  feet ; 
C.  -  W.  was  l'^  26'"  20«  at  6^^  a.m.  Sept.  20,  and  I''  27'"  11^  at  9''  26'" 
A.M.  of  the  same  date.  The  chronometer  had  been  compared  with 
a  standard  ashore,  and  found  to  be  fast  of  G.  M.  T.  0"*  26«  on 
Sept.  1  at  10  A.M.,  and  slow  of  G.  M.  T.  0'"  18«  on  Sept.  15  at  4  p.m. 
The  1918  almanac  gives  : 

Sept.  19,  20^^  G.  M.  T.,  decl.,  +  1°  22'.4 ;  equation,  +  6'"  17«.2. 

Sept.  19,  22^^  G.  M.  T.,  decl.,  +  1°  20'.5;  equation,  +  6'"  19^0. 

Sept.  20,    O''  G.  M.  T.,  decl.,  +  1°  18'.6;  equation,  +  6'"  20*.7. 

Sept.  20,    2"  G.  M.  T.,  decl.,  +  1°  16'.6;  equation,  +  6-  22«.5. 

Find  the  longitude  of  the  ship  by  the  time-sight  method.  Ans.  At 
the  time  of  observation  C.  —  W.  was  I''  26"*  49*.4 ;  chronometer 
was  slow  0'"  32«.4 ;  the  observation  being  a  forenoon  one,  the 
G.  M.  T.  came  out  21'»  25'"  48*  of  the  19th  Sept.  (p.  78) ;  by  formula 
(4),  p.  100,  hav.  (24^*  -  T)  was  9.38260;  corresponding  24"  -  T  was 
3'»55'"23«  (p.  264),  and  T  was  20'»4»»37«  (p.  103,  footnote) ;  ship's 
longitude  was  21°  52'  W. 

Ex.  6.  Simultaneously  with  the  altitude  measured  in  Ex.  5,  the 
sun's  compass  bearing  was  taken  with  a  pelorus  and  found  to  be 
123°.  The  variation  was  22°  W.,  by  the  magnetic  chart.  Find  the 
deviation.     Ans.    11°  E. 

This  example  may  be  solved  with  Table  11  because  the 
altitude  has  been  measured. 

Ex.  7.  Using  the  data  of  Ex.  5,  find  the  ship's  noon  latitude 
on  Sept.  20,  1918,  from  a  measured  noon  altitude  of  45°  46', 
Ans.    45°  18'. 

Ex.  8.     Calculate  Ex.  5  as  a  Sumner  by  the  Kelvin  Table. 

1  This  would  be  237°  -  180°  if  the  ship's  latitude  had  come  out 
greater  than  that  of  the  Sumner  point. 


MISCELLANEOUS   EXAMPLES  339 

Ans.  The  Sumner  point  is  in  latitude  45°  33' ;  longitude  21°  49' ; 
bearing  of  the  line  22°  or  180°  +  22°,  according  to  the  end  of  the 
line  to  be  used. 

Ex.  9.  From  the  noon  latitude  of  Ex.  7,  and  the  Sumner  line 
of  Ex.  8,  find  the  ship's  noon  longitude,  assuming  the  ship  was 
steaming  at  17  knots  on  a  168°  true  course.     Ans.  21°  2'. 

Ex.  10.  At  sea,  from  an  observation  at  S''  28"*  a.m.,  ship's  ap- 
parent time,  a  Sumner  point  was  computed  to  be  in  latitude  28°  26' 
N. ;  longitude  40°  11'  W. ;  bearing  of  the  line  28°  or  208°.  Clouds 
having  prevented  observation  at  noon,  the  latitude  was  found 
from  an  ex-meridian  observation  to  be  27°  17'  at  12^*  28"*  p.m.,  ship's 
time.  The  ship  was  steaming  at  18  knots  on  a  130°  true  course. 
Find  the  noon  latitude  and  longitude.  Ans.  Latitude,  27°  22' ; 
longitude,  39°  30'. 

Ex.  11.  With  the  data  of  Ex.  1,  it  is  required  to  prepare  in 
advance  for  an  ex-meridian  observation  and  its  calculation. 

Since  it  is  intended  to  make  the  observation  at  about  IP 
40"",  ship's  time,  we  begin  our  preparatory  calculations  by 
computing  K2  and  K3  for  ll''  SO'"  and  11^  44"*/  ship's  time, 
which  correspond  to  11''  36"*  44*  and  U'*  44"*  44'  by  the 
watch  2    We  thus  obtain  : 

for  IP  36"*  44%  declination  correction  =   —  28',   to  be  subtracted; 

alt.  correction  =  alt.  —  26°  50',  to  be  subtracted. 
for  11^  44"*  44*,  declination  correction  =  +  28',   to  be  subtracted; 

alt.  correction  =  alt.  —  27°  56',  to  be  subtracted. 

This  completes  the  preparatory  calculation.  In  Ex.  1 
the  actual  observation  of  altitude  was  made  at  IP  42"*,  and 
the  corrected  altitude  was  27°  8'.  Interpolating  the  declina- 
tion and  altitude  corrections  for  ll'*  42"*,  we  obtain : 

decHnation  correction  =   +  9' ;    alt.  correction  =  27°  8'  —  27°  34' 
=   -  26' ; 

both  corrections  to  be  subtracted.  We  then  have,  finally : 
Latitude  =  39°  -  9'  +  26'  =  39°  17'.  In  Ex.  1  we  found 
39°  20',  and  on  p.  149,  39°  19'. 

^  We  have  chosen  36"*  and  44"*  so  as  to  have  61  an  exact  number 
of  degrees.  This  increases  the  accuracy  of  Ki  (cf.  Ex.  1,  p.  336, 
footnote). 

2  We  know  from  the  data  of  Ex.  1  that  the  watch  was  44*  fast 
of  ship's  apparent  time. 


340  NAVIGATION 

Ex.  12.     With  the  data  of  Ex.  3,  prepare  in  advance  for  the 
calculation.     Ans.  We  find  : 
for  ll''  40"*,  declination  correction,  —  25',  to  be  added, 

alt.  correction  =  alt.  —  71°  20',  to  be  added; 
for  11^  48"*,  declination  correction,  —  28',  to  be  added, 

alt.  correction  =  alt.  —  71°  46',  to  be  added ; 
and  for  the  final  latitude  2°  52'.      In  Ex.  3  we  found  2°  53' ;    but 
such  small  differences  are  not  of  much  importance  in  navigation 
calculations. 

Ex.  13.  Using  the  data  of  Ex.  5  and  Ex.  9,  prepare  in  advance 
for  the  noon-sight  of  Ex.  7,  and  its  speedy  calculation. 

Ans.  D.  R.  longitude  at  noon,  21°  20' ;  watch  time  of 
noon,  IP  50"^  37*;  declination  at  noon,  +  1°  17';  D.  R. 
latitude  at  noon,  44°  20' ;  formula  (p.  89),  lat.  =  90°  +  dec. 
—  alt.  To  get  the  approximate  noon  altitude  in  advance, 
we  invert  the  formula,  and  thus  obtain  an  approximate 
'^D.  R.  alt."  =  90°  +  dec.  -  D.  R.  lat.  =  90°  +  1°  17'  - 
44°  20'  =  46°  57'.  For  this  D.  R.  alt.  at  noon,  we  find  that 
Table  6  +  Table  7  =  +  10'.  Therefore,  at  noon,  lat.  =  90° 
+  dec.  —  10'  —  observed  alt.  —  index  correction,  or  noon 
lat.  =  91°  4'  -  observed  alt.  =  91°  4'  -  45°  46'  =  45°  18'. 
This  number  (91°  4')  is  often  called  the  ''constant."  If  it 
has  been  prepared  in  advance,  the  latitude  can  be  calculated 
in  a  few  moments,  after  the  noon  observation  has  been  made 
at  about  U''  50""  37"  by  the  watch. 

Ex.  14.  With  declination  -  3°  7' ;  D.  R.  noon  latitude  +  38°  17' ; 
prepare  a  constant  for  a  noon-sight,  and  calculate  the  latitude,  sup- 
posing that  the  observed  altitude  turned  out  to  be  48°  17',  height 
of  eye  20  feet,  and  index  correction  +  3'.  Ans.  D.  R.  altitude, 
48°  36' ;   lat.  =  86°  39'  -  obs'd  alt.  =  38°  22'. 

Ex.  15.  With  the  data  of  Ex.  13,  and  at  11*  30'"  by  the  watch, 
it  is  required  to  set  it  so  that  it  will  be  correct  at  noon. 

Ans.  Move  the  hands  forward  from  IP  30"^  to  IP  39'" 
23 %  as  nearly  as  may  be  conveniently  possible.  (The  second 
hand  of  a  watch  should  always  be  set  so  as  to  be  on  60'  when 
the  minute  hand  is  exactly  on  one  of  the  minute  divisions 
of  the  dial.) 


MISCELLANEOUS   EXAMPLES  341 

Ex.  16.  Prepare  a  constant  for  a  meridian  observation  of  jS 
Cassiopeiae,  Dec.  20,  1917,  and  determine  in  advance  the  approxi- 
mate time  for  the  observation.  D.  R.  latitude,  39°  18'  N.,  D.  R. 
longitude,  33°  l'  W.,  both  calculated  for  8  p.m.  ;  ship  steaming  11 
knots  due  E.  by  compass  ;  variation,  24°  W. ;  deviation,  3°  E.  Also 
calculate  the  latitude,  supposing  the  observed  altitude  turned  out  to 
be  70°  54',  with  eye  20  feet  and  index  +  3'.  Ans.  Ship's  time  of 
observation,  G''  ll'"  p.m.;  lat.  =  obs'd  alt  -  31°  19'  =  70°  54'  - 
31°  19'  =  39°  35'.     The  constant  is  31°  19'. 

Ex.  17.  On  the  ship  of  Ex.  16,  Dec.  20,  1917,  at  6^^  38"*  23«  p.m. 
by  the  watch,  the  altitude  of  Aldebaran  or  a  Tauri  was  measured, 
and  found  to  be  33°  25'.  C.  -  W.  was  2^  12"^  48« ;  chron.  fast 
2"*  26*.  Find  the  longitude,  using  a  D.  R.  latitude ;  and  also  run  a 
Sumner  line.  (Note.  The  correction  for  "time  past  noon"  in  this 
example  is  1"*  27*. )  Ans.  Longitude,  33°  13'  W. ;  Sumner  point,  lati- 
tude, 39°  15' ;  longitude,  33°  13' ;  bearing  of  the  line,  6°  or  180°  +  6°. 

Ex.  18.  From  the  Sumner  Hne  of  Ex.  17  and  the  latitude  of  Ex. 
16  find  the  longitude  at  6''  ll"*,  when  the  meridian  observation  was 
made.     Ans.    33°  16'.    . 

Ex.  19.  A  ship  is  to  proceed  (p.  19)  from  Sandy  Hook  (lat., 
40°  28'  N.;  long.,  73°  50'  W.)  to  St.  Vincent  (lat.,  16°  50'  N. ; 
long.,  25°  7'  W.).  A  straight  Une  being  drawn  between  these  two 
points  on  the  North  Atlantic  great  circle  sailing  (or  gnomonic) 
chart  (p.  38),  it  was  found  to  cross  the  successive  principal  longi- 
tude meridians  at  the  following  points : 

A,  lat.,  39°  37' ;  long.,  70°  O' ;  B,  lat.,  36°  39' ;  long.,  60°  O' ; 
C,  lat.,  32°  34';  long.,  50°  O' ;  D,  lat.,  27°  lO' ;  long.,  40°  O'; 
E,  lat.,  20°  30';  long.,  30°  O'. 

The  shortest  track  between  Sandy  Hook  and  St.  Vincent  will 
therefore  pass  through  these  successive  points  (see  p.  38).  It  is 
required  to  calculate  logarithmically,  by  middle  latitude  saiUng 
(p.  35),  the  successive  courses  and  distances  between  these  points, 
so  as  to  compare  them  with  the  middle  latitude  course  and  distance 
from  Sandy  Hook  to  St.  Vincent  direct.  The  middle  latitude  is  to 
be  taken  to  the  nearest  minute  in  each  case.     Ans. 

Course  Dist. 

Sandy  Hook  to  A          106°     9'  183.3 

Aio  B          110°  40'  504.4 

B  to  C          116°  23'  551.3 

C  to  D          121°  55'  613.0 

DioE          126°     5'  679.1 

E  to  St.  Vincent          128°  24'  354.2 

Total  distance  by  great  circle  saihng  2885.3 


342  NAVIGATION 

Middle  latitude  sailing,  Sandy  Hook  to  St. 
Vincent  direct, 

course,  118°  56'  dist.  .        2931.0 

'  Apparent  saving  of  distance  by  great 

circle  sailing,  45,7 

It  will  thus  be  seen  that  the  great  circle  course  on  leaving 
the  Hook  is  more  than  a  whole  compass  point  to  the  north- 
ward of  the  middle  latitude  course,  being  106°  9',  instead  of 
118°  56'. 

Ex.  20.  A  sub-chaser  with  a  cruising  speed  of  12  knots  is  bound 
from  Norfolk  to  New  York.  While  on  the  way,  the  navigator  is 
required  to  find  her  true  course  and  distance  from  a  point  off  Winter 
Quarter  Lightship  (lat.,  37°  54' ;  long.,  74°  54'),  to  a  point  off  N.  E. 
End  Lightship  (lat.,  38°  56' ;  long.,  74°  27'),  assuming  that  a  ^-knot 
flood-current  set  into  the  mouth  of  the  Delaware  in  a  N.  W.  direc- 
tion during  3  hours  of  the  run. 

Ans.  If  the  chaser  shaped  her  course  without  regard  to 
the  tidal  current,  she  would,  after  running  down  her  dis- 
tance, be  1 J  miles  N.  W.  of  her  intended  destination  off  N.  E. 
End  ship.  To  avoid  this,  her  course  should  be  shaped  for 
a  point  1§  miles  S.E.  of  her  intended  destination,  and  then  the 
current  will  cause  her  to  reach  the  original  desired  point. 
The  easiest  way  to  make  the  calculation  is  to  use  the  method 
of  traverse  sailing  (p.  39).  This  requires  that  we  calculate 
the  latitude  difference  and  departure,  separately,  both  for 
the  ship's  run  and  for  the  current,  and  then  correct  the 
former  with  the  latter  before  taking  from  the  traverse  table 
the  ship's  final  course  and  distance.  We  first  calculate  for 
the  run  from  Winter  Quarter  to  N.  E.  End,  using  the  lati- 
tudes and  longitudes  given  above,  and  obtain : 

For  ship's  run  without  Lat.  Diff.             Dep. 

regarding  current     ....  62.0,  northerly ;  21.2,  easterly; 

1|  miles,  N.W.  current  ....  1.0,  northerly;     1.0, westerly; 

Subtracting  the  current  effect      .  61.0,  northerly;  22.2,  easterly; 

and  corresponding  to  latitude  difference  61.0,  departure  22.2, 
the  Traverse  Table  gives  true  course  for  the  ship  20°,  dis- 
tance 65  miles.  The  course  without  regard  to  current  would 
have  been  19°. 


INDEX 


Accuracy,   how  attained  in  Sumner    Argument,  difference,  12,  27 


method,  137 

in  interpolating  declination,  82 

in  reading  sextant,  66,  73 
Address,  of  house  in  city,  2 

of  ship  at  sea,  3 
Adjuster,  compass,  43,  323 

deviation  table,  48 
Adjusting  screws,  68 
Adjustments  of  sextant,  68 
Advance    preparation    for    observa- 
tion, star,  96 

sun,  92 
"Afternoon"     observation    of    star, 

120 
Aid  to  navigation,  5. 
Almanac,  nautical,  chapter  on,  75 

specimen  pages,  76,  83,  84,  91,  92, 
97,  98 
Altitude,  changes  slowly  near   noon, 
88 

defined,  61 

maximum  and  minimum,  89 

maximum  of  star,  90 

sun  or  star,  correction  of,  70 

used  for  noon-sight,  86 
A.M.  and  P.M.,  75 
American  ephemeris,  75 
Angle,  auxiliary,  in  Sumner  method, 
112 

auxiliary,  in  azimuth  table,  115 

course,  10 

danger,  58 

deck,  59 

defined,  8 

how  designated,  8 

how  measured,  9 

of  triangle,  31 

pairs  of,  for  bow  bearings,  56 

vertex  of,  8 
Apparent  solar  day,  77 

time,  77 

time,  used  for  ship's  clock,  94 
Arc,  minutes  of,  3 


double,  in  haversine  table,  100 

in  tables,  10 

pairs  of,  11 

pairs  of,  in  log  table,  25 

pairs  of,  in  traverse  table,  14 

pairs    of,    in    trigonometric    table, 
32 
Arrow  on  sextant  vernier,  65 
Artificial  horizon,  69 

accuracy  of,  73 
Ashore,  observing  with  sextant,  69 
Astronomic  time,  75,  78 
Auxiliary,    angle,    in   azimuth   table, 
115 

angle,  in  Sumner  method,  112 

azimuth  table,  115 
Avenues    and    streets     imagined    by 

navigator,  3 
Azimuth,   circle   (instrument),  44,  53 

how  counted,  114 

of  sun,  defined,  44 

table.  111,  113 

table,  auxiliary,  115 

Beam,  bearing,  55 
Bearing,  bow  and  beam,  66 

compass,  of  sun,  44 

cross,  56 

danger,  59 

favorable,  for  time-sight,  99 

from  the  bow,  doubling  it,  54 

from  the  bow,  in  Sumner  naviga- 
tion, 134 

of  objects  ashore,  53 

of  Sumner  line.  111 

reversed,  55 
Below  pole  observations,  89 
Bow,  and  beam  bearings,  55 

bearings  from  the,  doubling,  54 

bearings  from  the,  use  in  Sumner 
navigation,  134 
Bowl  of  compass,  42 
Boxing  the  compass,  41 


343 


344 


INDEX 


Cabin  clock,  how  set,  94 

Calipers,  used  in  testing  sextant,  69 

Card,  compass,  41 

rate,  of  chronometer,  79 
Cardinal  points  of  compass,  41 
Celestial,  equator,  85 

Greenwich,  85 

poles,  85 

sphere,  85 
Changed    D.    R.    point    in    Sumner 

navigation,  126 
Chart,  great  circle  sailing,  37 

how  to  draw  Sumner  lines  on,   123 

in  dead  reckoning,  8 

laying  down  ship's  place  on,  55 

Mercator,  35 

Mercator,  meridians  on,  38 

projections,  38 

scale  of,  55 
Chronometer,  6 

dial,  divided  in  12  hours,  93 

error,  79 

face,  93 

keeps  mean  solar  time,  77 

minus  watch,  94 

rated,  79 

used  in  interpolating  declination, 
90 
Circle,  graduated,  of  sextant,  61 
Civil  day,  77 

time,  78,  93 
Clamp  of  sextant,  62 
Clock,  deck  and  cabin,  how  set,  94 
Clockwise,  numbering  of  compass,  41 
Coastwise  navigation,  chapter  on,  53 
Compass,  adjuster,  43,  323 

bearing  of  objects  ashore,  53 

bearing  of  sun,  44 

bowl,  42 

boxing,  41 

card,  41 

chapter  on,  40 

compensating,  43 

course  and  true  course,  42 

deviation  by  observation,  44,  115 

deviation  table,  48 

error  of,  43 

error  of,  when  negative,  47 

formulas,  45 

gyro,  42 

lubber  line,  42 

north,  43 

old-fashioned  designation  of  error, 
47 


Compass,  points  of,  10,  15,  52 

points,  cardinal  and  inter-cardinal, 
41 

variation,  43 
Compensating  compass,  43 
Composite  sailing,  39 
Correction,  of  altitudes,  70 

tables,  for  altitudes,  72 
Cosecant,  31 
Cosine,  31 

-haversine    formulas,    in    Sumner 
method,  112 
Cotangent,  31 
Course,  angle,  10 

magnetic,  49 

protractor,  55 

ship's,  defined,  8 

ship's,  error  of  compass,  43 

ship's,  in  points,  15 

ship's,  measured,  9,  19 

ship's,  old  way  of  measuring,   19, 
22 

ship's,  shaping  it,  44 

ship's,  true  and  compass,  42 

steering,  51 
Cross-bearings,  56 
Currents,  ocean,  effect  of,  2 

ocean,  measured,  103 

Daily  rate  of  chronometer,  79 
Danger,  angle,  57 

bearing,  59 
Date,  right,  in  almanac,  81 
Day,  apparent  solar,  77 

at  sea,  navigator's,  141 

civil,  77 

mean  solar,  77 

solar,  75 
Dead  reckoning,  begins,  53 

chapter  on,  7,  23 

defined,  5 

fundamental  problems  of,  8,  10,  20 

point,  changed,  in  Sumner  naviga- 
tion, 126 

point,  in  Sumner  navigation,  112 

with  logs,  23,  31,  33 
Decimal  part  of  log,  23 

rule  for,  25 
Deck,  angle,  59 

clock,  how  set,  94 
Declination,     interpolated     without 
chronometer,  90 

north  and  south,  plus  and  minus, 
89 


INDEX 


345 


Declination,  of  star,  85,  89,  92 

of  star,  slow  change  of,  90 

of  sun,  75,  82,  85 

of  sun,  minus,  87 
Degrees,  of  latitude    and  longitude, 
when  equal,  4 

of  longitude,  3 

of   longitude,    counted    East    and 
West,  3 

on  compass  card,  41 

subdivisions  of,  3 

unit  for  measuring  angles,  9 
Departure,  defined,  10 

new,  2 

port  of,  1 

relation  of  longitude  difference  to, 
16,  20 

summed,  39 

taking  a,  53 
Destination,  port  of,  1 
Deviation,  of  compass,  43 

of  compass,  by  observation,  44,  115 

of  compass,  older  designation  of,  47 

of  compass,  when  negative,  47 

table,  48,  49 

table,  inverse  use  of,  48 

table,  second,  49 
Dial,    chronometer,    divided    in    12 
hours,  93 

of  compass,  41 
Difference,  argument,  12,  27 

hourly,  79 

interpolation,  28 

latitude,  defined,  10 

latitude,  in  miles  and  minutes,  15 

latitude,  meridional,  36 

latitude,  summed,  39 

longitude,    converted   into    depar- 
ture, 15,  19 

tabular,  12,  27 

time,  80 
Digits,  defined,  24 
Dip,  of  sea  horizon,  70 

of  sea  horizon,  errors  in,  73 

of  sea  horizon,  table  of,  73 

of  star,  at  upper  transit,  96 

of  sun,  at  noon-sight,  86 
Direction,  defined,  8 

difference  of,  8 
Disk  of  sun  in  telescope,  67 
Distance,  in  dead  reckoning,  10 

in  traverse  table,  13 

laying  off  on  chart,  55 

polar,  100 


Dividend,  log  of,  30 

Division,  by  logs,  30 

Divisor,  log  of,  30 

Documents,  superintendent  of,  75 

Double,  argument  in  haversine  table, 

100 
interpolation,  12 
Doubling  the  bearing  from  the  bow, 

54 

Ephemeris,  American,  75 
Equation  of  time,  75,  77 
Equator,  celestial,  85 

earth's,  a  circle,  4 

longitude  meridians  begin  at,  3 
Equinox,  vernal,  85 
Error,  index,  of  sextant,  66 

in  tabular  dip  of  horizon,  73 

of  chronometer,  79 

of  compass,  43 

of  compass,  older  designation  of,  47 

of  compass,  when  negative,  47 
Exactness  of  logs,  30 
Ex-meridian  observation,  99 

Face,  chronometer,  93 

Factor,  correction,  in  Table  2,  17 

log  of,  30 
Favorable  bearing  for  time-sight,  99 
"Fix"  defined,  53 

"Forenoon"  observation  of  star,  120 
Formulas,   cosine-haversine,  for  alti- 
tude, 112 

for  compass,  45 

for  dead  reckoning,  33,  34 

for  latitude  by  noon-sight,  87,  89 

for  magnetic  course,  49 

for  Mercator  sailing,  36 

for  operations  by  Kelvin's  Sumner 
table,  127 

for  Sumner  intersection  point,  135 

for  time-sight,  100 

for  time-sight,  star,  105 

Gaining  rate  of  chronometer,  79 
Graduated  circle,  of  sextant,  61 

off  arc,  66 
Great  circle,  37 

chart,  38 

sailing,  35,  37 
Greenwich,  beginning  of  longitude,  3 

celestial,  85 

mean  time,  77 
Gyro-compass,  42 


346 


INDEX 


Half  sum,  100 

Haversines,  99 

High  latitudes,  noon-sight  in,  89 

Horizon,  61 

artificial,  69 

dip  of,  70 

glass,  on  sextant,  64 
Hour-angle  of  star,  104,  120 
Hourly  difference,  79 

Imaginary  ship,   in   Sumner  naviga- 
tion, 134 
Index,  error,  of  sextant,  66 

error,  of  sextant,  from  sun  obser- 
vations, 67 

glass,  of  sextant,  64 

number,  in  azimuth  table,  114 

of  sextant,  62 
Initial  meridian,  Greenwich,  3 
Inter-cardinal  points  of  compass,  41 
Interpolation,  12 

difference,  28 

double,  12 

in  trigonometric  tables,  33 

inverse,  29 

of  logs,  27 
Intersection  point  of   Sumner  lines, 
111,  125 

finding  it,  133 
Inverse  interpolation,  29 

use  of  deviation  table,  48 

use  of  mathematical  tables,  12 

use  of  Table  3,  28 

use  of  Table  4,  33 
Inverting  telescope,  72 

Kelvin,  Lord,  his  table,  126 

improves  Sumner's  method,  112 

Land-marks,  observed  from  ship;  53 
Lane  route,  95 

Latitude,  degrees  of,  equal  to  degrees 
of  longitude,  4 

determined  by  noon-sight,  85 

determined  by  sextant,  5 

determined  by  star,  89 

difference,  defined,  10 

difference,  in  miles  and  minutes,  15 

difference,  meridional,  36 

difference,  summed,  39 

for  time-sight  calculation,  101 

high,  noon-sight  in,  89 

parallels  of,  defined,  3 


Laying  down  ship's  place  on  chart,  55 

off  distance  on  chart,  55 
Lead-line,  60 
Light-gathering  power    of  telescope, 

63 
Lighthouse,  aid  to  navigation,  5 

observed  from  ship,  53 
Limb,  lower,  of  sun,  71 
Line  of  soundings,  60 

Sumner,  or  line  of  position,  109 

Sumner,  principle  of.  111 
Log,  abbreviation  for  logarithm,  23 

instrument  for  estimating  speed  of 
ship,  54 
Logarithms,     decimal   part   of,    rule 
for,  25 

exactness  of,  30 

in  dead  reckoning,  chapter  on,  23 

interpolation  of,  27 

multiplication   and   division   with, 
30 

of  multiples  of  10,  24 

of  products,  30 

table  of,  25 

table  of,  inverse  use  of,  28 

table  of,  trigonometric,  32 

two  parts  of,  23 

whole  number  part  of,  rule  for,  24 
Longitude,  begins  at  Greenwich,  3 

degrees  of,  3 

degrees  of,  counted  east  and  west, 
3 

degrees  of,  when  equal  to  degrees 
of  latitude,  4 

determined  from  star,  104 

determined  from  sun,  99 

difference,   converted    into    depar- 
ture, 15,  19 

difference,    converted    into    time- 
difference,  81 

meridians  of,  defined,  3 

minutes  and  seconds  of,  4 

not  determinable  by  noon-sight,  88 
Losing  rate  of  chronometer,  79 
Lower,  limb  of  sun,  71,  86 

transit,  89 
Lubber-line  of  compass,  42 

Magnetic  course,  49 
Maps,  lines  on,  4 
Mathematical  tables,  10 

inverse  use  of,  12 
Maximum,  altitude,  89 

of  star,  90 


I 


INDEX 


347 


Mean,  of  observations,  73 

solar  day,  77 

solar  time,  77 

sun,  83 

time,  Greenwich,  77 
Mercator,  charts,  35 

charts,  meridians  on,  38 

Gerhard,  35 

sailing,  35 
Mercury,  in  artificial  horizon,  69 
Meridian,  how  numbered,  3 

initial,  at  Greenwich,  3 

of  longitude,  defined,  3 

on  maps,  4 

on  Mercator  chart,  35 

observation,  86 

transit,  96 
Meridional,  latitude  difference,  36 

parts,  35 
Middle  latitude,  16 

sailing,  35 

sailing,  formulas  for,  33,  34 
Midnight  sun,  observed  for  latitude, 

89 
Miles,  nautical,  15 
Minimum,  altitude,  89 
Minus,  declination,  82,  87 

defined,  25 

sign,  for  south  latitude  and  decli- 
nation, 89 

sign,  in  formula,  87 
Minutes  of  arc,  3 

subdivided,  4 

used  in  measuring  angles,  9 
Mirror,  artificial  horizon,  69 

sextant,  61 
Moon,  observation  of,  99 
Motion,  of  ship,  between  two  Sum- 
ner observations,  137 
Multiples  of  10,  logs  of,  24 
Multiplication,  by  logarithms,  30 

table,  11,  28 

Nautical  almanac,  chapter  on,  75 

miles,  15 

specimen  pages,  76,  83,  84,  91,  92, 
97,  98 
Navigator,  his  day  at  sea,  141 
Negative,  compass  error,  47 

defined,  24 
Newer  navigation  methods,  chapter 

on,  108 
Noon,  75 

sight,  86 


Noon,  sight,  advance  preparation  for, 
92 

sight,  good  for  latitude  only,  88 

sight,  in  high  latitudes,  89 

sight,  in  tropics,  89 

sight,  of  star,  90 
North,  by  compass,  43 

declination  and  latitude  are  plus, 
89' 
Numbers,  of  longitude  meridians,  3 

tabular,  10 

Objections  to  Sumner  line,  110" 
Observation,     advance     preparation 
for  sun,  92 
advance  preparation  for  star,  96 
determines  ship's  position,  4 
for  compass  deviation,  44 
planets  and  moon,  98 
single,  determines  one  thing  only, 
88 
Ocean  currents,  deflect  ship,  2 

measured,  103 
Off  arc,  graduations  of  sextant,  66 
Older    navigation    methods,    chapter 
on,  86 

Pages  from  nautical  almanac,  76,  83, 

84,  91,  92,  97,  98 
Pairs,  of  angles  for  bow  bearings,  56 

of  arguments,  11 

of  arguments,  in  log  table,  25 

of  arguments,  in  traverse  table,  14 
Parallax,  correction  of  sun's  altitude, 

71 
Parallel,  of  latitude,  defined,  3 

of  latitude,  how  counted,  3 

of  latitude,  on  maps,  4 

rulers,  55 

sailing,  39 
Parts,  meridional,  35 

of  triangle,  31 

proportional,  28 

two,  of  compass  error,  43 

two,  of  log,  23 

two,  of  log,  rule  for,  24 
Patent  log,  54 
Pelorus,  44,  53 
Plane,  defined,  9 

sailing,  9 

trigonometry,  10 
Planets,  observation  of,  for  latitude, 
98 

observation  of,  time-sight,  104 


348 


INDEX 


Plotting  ship's  place  on  chart,  55 
Plus,  sign  in  formula,  87 

sign,    north    declination    and    lati- 
tude, 89 
P.M.  and  A.M.,  75 

Point,   D.  R.,  in  Sumner  navigation, 
112 

intersection,  of  Sumner  lines.  111, 
125 

Sumner,  112 

Sumner,  ship  not  at,  125 
Points  of  compass,  10,  15 

cardinal,  41 

inter-cardinal,  41 

table  of,  52 
Polar  distance,  100 
Pole,  celestial,  85 

celestial,  observing  below  it,  89 

end  of  longitude  meridians,  3 
Port  of  departure  and  destination,  1 
Position,  line  of,  109 

of  ship  at  sea,  defined,  2 

of  ship,  determined  daily,  2 

of  ship,  how  determined,  4 

of    ship,    most    probable,    not    at 
Sumner  point,  125 
Preparation  for  observation,  star,  96 

sun,  92 
Principle  of  Sumner  line,  111 
Probable    position    of    ship,    not    at 

Sumner  point,  125 
Problem,   of   dead   reckoning,   8,   10, 
20 

of  dead  reckoning,  with  logarithms, 
23,  31,  33 

of  navigation,  1 
Product,  log  of,  30 
Projection  chart,  38 
Proportional  parts,  28 
Protractor,  instrument,  55,  58 

Quotient,  log  of,  30 

Rate  of  chronometer,  79 
Reading  sextant  circle,  62 
Refraction  of  light,  71 
Reversed  bearing,  55 
Rhumb  line,  38 
Right-ascension,  of  star,  85,  89,  91 

of  star,  slow  change  of,  90 

of  sun,  83 
Roof,  of  artificial  horizon,  69 
Rulers,  parallel,  55 


Sailing,  composite,  39 

Mercator,  35 

middle  latitude,  35 

parallel,  39 

plane,  9 

traverse,  39 
Saint      Hilaire,      Marcq,      improves 

Sumner's  method,  112 
Scale  of  chart,  55 
Screw  adjusting,  68 
Secant,  31 

Second  deviation  table,  49 
Seconds  of  arc,  4 
Semi-diameter  of  sun,  71 
Set  of  current  measured,  103 
Setting  watch  on  board,  94 
Sextant,  adjustments,  68 

chapter  on,  61 

clamp,  62 

defined,  5 

determines  latitude,  5 

for  danger  angle,  59 

graduated  circle  of,  61 

graduations  off  arc,  66 

horizon  glass  of,  64 

index,  62 

index  glass,  64 

mirrors,  62 

read  to  minutes  only,  66,  73 

superposed  objects,  64 

tangency  of  sun's  image,  67,  70 

tangent  screw,  62 

telescope,  62 

use  of,  ashore,  69 

vernier,  64 
Shaping  a  course,  39,  44 
Ship,  course,  defined,  8 

course,  measured,  9,  19 

course,  measured,  old  way,  19 

imaginary,  in  Sumner  navigation, 
134 

motion   of,    between   two   Sumner 
observations,  137 

position  of,  how  determined,  4 

position  of,  laying  down  on  chart, 
55 

swinging,  for  compass  adjustment, 
43 

time  is  apparent  solar  time,  94 
Sidereal  time,  85 
Sides  of  triangle,  31 
Sine,  31 

Single   observation     determines   one 
thing  only,  88,  111,  125 


INDEX 


349 


Sky,  85 

Slow-motion  screw  of  sextant,  62 

Solar  day,  75 

apparent,  77 

mean,  77 
Solar  time,  75 

apparent,  77 

mean,  77 
Soundings,  59 

machine,  60 
South      declination      and      latitude, 

minus,  89 
Specimen      pages,      from      nautical 
almanac,  76,  83,  84,  91,  92,  97,  98 
Sphere,  celestial,  85 
Star,  advance  preparation  for  obser- 
vation, 96 

dip  of,  96 

"forenoon"  and  "afternoon"  ob- 
servation, 120 

hour-angle  of,  104 

"noon-sight"  of,  90 

observed  for  index  error,  67 

observed  for  latitude,  89 

observed  for  Sumner  line,  120 

observed  for  time-sight,  104 

right-ascension  and  declination,  85, 
91,  92 

time  of  upper  transit,  96 
Steering  courses,  table  of,  51 
Streets    and    avenues,    imagined    by 

navigator,  3 
Subdivisions,  of  degrees,  etc.,  3 

of  minutes,  4 
Summed  latitude  difference  and  de- 
parture in  traverse  sailing,  39 
Sumner,  Capt.  Thos.  H.,  109 
Sumner  intersection  table,  135 
Sumner  line,  109 

bearing  of.  111 

how  to  draw  it  on  chart,  123 

objections  to  it,  110 

point  of  intersection,  111,  125 

point  of  intersection,  finding  it,  133 

principle  of,  111 

star  observed  for,  120 

sun  observed  for,  112 
Sumner  method,  auxiliary  angles  in, 
112 

conditions  for  accuracy  in,  137 

cosine-haversine  formula,   112 

with  special  tables,  126 
Sumner   navigation,    compared   with 
time-sight,  124 


Sumner  navigation,  use  of  bearings 

from  the  bow  in,  134 
Sumner  observations,  motion  of  ship 

between  two,  137 
Sumner  point,  112 

ship  not  at,  125 
Sun,  advance  preparation  for  obser- 
vation of,  92 

altitude  correction  of,  70 

azimuth  of,  44 

compass  bearing  of,  44 

declination  of,  75,  82,  85 

in  nautical  almanac,  76 

its  disk  in  telescope,  67 

lower  limb  of,  71 

mean,  77,  83 

midnight,  observed  for  latitude,  89 

observed  for  index  error,  67 

observed  for  Sumner  line,  112 

right,  ascension  of,  83,  85. 
Superintendent  of  documents,  75 
Superposed  objects  in  sextant,  64 
Supplementary    correction    of    sun's 

altitude,  72 
Swinging  ship   for    compass    adjust- 
ment, 43 

Table,  auxiliary  azimuth,  115 

azimuth.  111 

compass  points,  52 

corrections  for  altitudes,  72 

deviation,  48 

deviation,  second,  49 

dip,  73 

haversines,  99 

logarithms,  25 

logarithms,  inverse  use  of,  28 

mathematical,  10 

mathematical,  inverse  use  of,  12 

multiplication,  11,  28 

of  steering-courses,  51 

Sumner,  by  Kehan,  126 

Sumner  intersection,  135 

traverse,  10 

traverse,  used  to  convert  longitude 
difference  into  departure,  19 

trigonometric  logarithms,  32 
Tabular,  difference,  12,  27 

numbers,  10 
Taking  a  departure,  53 
Tangency  of  sun's  images,  67,  70 
Tangent,  images  of  sun,  67,  70 

in  trigonometric  tables,  31 

screw  of  sextant,  62 


350 


INDEX 


Telescope,  inverting,  72 

of  sextant,  61 
Ten,  arbitrary  increase  of  logs  by,  25, 
31,  32 
multiples  of,  logs  of,  24 
Time,  apparent  solar,  77 

apparent    solar,    used    for    ship's 

clock,  94 
astronomical,  78 
civil,  78,  93 
difference,  80,  87 
equation  of,  75,  77 
Greenwich  mean,  75 
mean  solar,  77 
sidereal,  85 
solar,  75 

transformation,   mean  to  sidereal, 
85 
Time-sight,  99 

calculation,  latitude  for,  101 
compared  with  Sumner  navigation, 
124 
Transformations,    time,    importance 
of,  82 
mean  to  sidereal,  85 
Transit,  lower,  89 

upper,  89 
Traverse,  sailing,  39 
table,  10 

table,  in  Sumner  method,  113,  135 
table,    used    to    convert   longitude 
difference  and  departure,  19 


Triangle,  parts  of,  31 
Trigonometry,  log  table,  31 

plane,  10,  31 
Tropics,  noon-sight  in,  89 
True  course  of  ship,  42 

north,  43 

Unit  for  measuring  angles,  9 
Upper  transit,  89 
time  of,  for  stars,  96 

Variation  of  compass,  43 

older  designation  of,  47 

when  negative,  47 
Vernal  equinox,  85 
Vernier,  backward  reading,  off  arc,  66 

of  sextant,  64 
Vertex  of  angle  defined,  8 

Watch,  chronometer  minus,  94 

how  set,  94 

used  for  timing  observations,  94 
Whole  number  part  of  log,  23 

rule  for,  24 
Wind,  deflects  ship,  2 

Zenith,  defined,  61 

sun  near  it  in  tropics,  89 
Zero,  hours,  same  as  noon,  75 

meridian,  Greenwich,  3 


1 


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MACMILLAN  BOOKS  ON  ASTRONOMY 
(TEXT-BOOKS) 

Astronomy — A  Popular  Handbook 

By  HAROLD  JACOBY 

Rutherfurd  Professor  of  Astronomy  in  Columbia  University 

Decorated  cloth,  illustrated,  j2  plates 
and  many  figures  in  the  text,  8vo,  $2 .50 

This  book  has  been  made  to  meet  the  wishes  of  the  ordi- 
nary reader  who  may  desire  to  inform  himself  as  to  the 
present  state  of  astronomic  science  or  to  secure  a  simple 
explanation  of  the  many  phenomena  continually  exhibit- 
ing themselves  in  the  universe  about  him.  In  order  that 
the  non-scientific  reader  may  not  be  hampered  by  formulae 
and  symbols  of  which  he  knows  little,  Professor  Jacoby 
has  divided  the  book  into  two  parts,  the  first  being  free 
from  mathematics,  and  the  second  a  series  of  extended 
elementary  mathematical  notes  and  explanations  to  which 
appropriate  references  are  made  in  the  first  part.  The 
general  reader  may,  therefore,  confine  his  attention  to  the 
non-mathematical  part,  the  student  to  the  whole  volume. 
The  work  is  elaborately  illustrated,  there  being  32  full-page 
plates  from  photographs  and  many  text  figures. 

"It  serves  Its  purpose  better  than  any  recent  hand- 
book, and  ought  to  lead  many  Inquirers  toward  this 
fascinating  study." — New  York  Times, 

"The  author  has  the  gift  of  making  technical  sub- 
jects seem  easy,  and  In  this  book  he  has  Incorporated 
all  the  latest  results  of  the  science  of  the  heavens." 

— San  Francisco  Chronicle, 


THE  MACMILLAN  COMPANY 

PubllBhers        64-66  Fifth  Avenue        New  York 


An  Introduction  to  Celestial  Mechanics 
By  F.  R.  MOULTON 

Professor  of  Astronomy  in  the  University  of  Chicago 

Second  edition,  revised,  diagrams,  8vo,  $3.75 

Intended  to  give  a  satisfactory  account  of  many  parts 
of  celestial  mechanics  rather  than  an  exhaustive  treatment 
of  any  special  part ;  to  present  the  work  so  as  to  attain 
logical  sequence,  to  make  it  progressively  more  difficult, 
and  to  give  the  various  subjects  the  relative  prominence 
which  their  scientific  and  educational  importance  deserves. 
In  short,  the  aim  has  been  to  prepare  such  a  book  that 
one  who  has  had  the  necessary  mathematical  training  may 
obtain  from  it,  in  a  relatively  short  time  and  by  the  easiest 
steps,  a  broad  and  just  view  of  the  whole  subject. 

"  Composed  with  remarkable  good  judgment,  and  indis- 
pensable to  all  students  of  the  subject."  —  N.  Y.  Post. 


THE   MACMILLAN   COMPANY 

PubliBhers  64-6$  Fifth  Aveau9  New  York 


MACMILLAN  BOOKS  ON  ASTRONOMY 
(TEXT-BOOKS) 

By  forest  ray  MOULTON,   Ph.D. 

Assistant  Professor  of  Astronomy  in  the  University  of  Chicago 

An  Introduction  to  Astronomy 

Revised  edition,  III.,  cloth,  8vo,  $2.40 

This  book  gives  an  introductory  account  of  the  present  state 
of  the  science  of  astronomy,  logically  arranged  and  written,  so 
that  it  may  be  easily  comprehended  by  the  student  without  math- 
ematical or  extensive  scientific  training,  and  so  that  he  may 
obtain  from  it  not  only  some  knowledge  of  scientific  achieve- 
ments, but  also  some  of  the  spirit  which  inspires  scientific  work. 
To  secure  the  benefits  of  the  laboratory  method  numerous  sug- 
gestions and  exercises  for  practical  observations,  both  with  and 
without  a  telescope,  are  given.  Finally,  the  aim  has  been  to 
give  the  student  a  well-balanced  conception  of  the  astronomy  of 
the  present  day. 

The  Elements  of  Practical  Astronomy 

By  W.  W.  CAMPBELL 

Astronomer  in  the  Lick  Observatory 

Cloth,  8vo,  254  pages,  $2.25 

The  elements  of  practical  astronomy,  with  numerous  applica- 
tions to  the  problems  first  requiring  solution.  It  is  suited  for 
use  with  students  who  have  had  an  introductory  training  in 
astronomy  and  mathematics. 


THE   MACMILLAN   COMPANY 

Publishers  64-66  Fifth  Avenue  New  Tork 


MACMILLAN  BOOKS   ON  ASTRONOMY 
(TEXT-BOOKS) 

By  WILLIS   I.  MILHAM,   Ph.D. 

Field  Memorial  Professor  of  Astronomy  in  Williams  College 

How  to  Identify  the  Stars 

Cloth,  i2mo,  j8  pages,  75  cents 

The  purpose  of  this  little  book  is  to  serve  as  a  guide  in  tak- 
ing the  first  steps  in  learning  the  stars  and  constellations  and 
also  to  point  the  way  to  the  acquisition  of  further  information 
on  the  part  of  those  who  desire  it.  Excellent  star  maps  are 
included. 


Meteorology 

A  Text-book  of  the  Weather,  the  Causes 
OF  ITS  Changes,  and  Weather  Forecasting 
FOR    THE    Student    and    General    Reader 

Cloth,  8vo,  illustrated,  54g  pages,  $4.50 

This  book  is  essentially  a  text-book.  For  this  reason,  the 
marginal  comments  at  the  sides  of  the  pages,  the  questions, 
topics  for  investigation,  and  practical  exercises  have  been 
added.  A  syllabus  of  each  chapter  has  been  placed  at  its 
beginning,  and  the  book  has  been  divided  into  numbered  sec- 
tions, each  treating  a  definite  topic.  The  book  is  also  in- 
tended for  the  general  reader  of  scientific  tastes ;  for  while  it 
can  hardly  be  called  an  elementary  treatise,  it  starts  at  the 
beginning  and  no  previous  knowledge  of  meteorology  itself  is 
anywhere  assumed.  It  is  assumed,  however,  that  the  reader  is 
familiar  with  the  great  general  facts  of  science.  References 
have  been  added  at  the  end  of  each  chapter. 


THE   MACMILLAN   COMPANY 

Publishers  64-66  Fifth  Avenue  New  Tork 


•v 


MACMILLAN  BOOKS   ON  ASTRONOMY 
(TEXT-BOOKS) 

By  WILLIS   I.  MILHAM,   Pf 

Field  Memorial  Professor  of  Astronomy  in  Wil' 


How  to  Identify  the  Stars 


The  purpose  c^  +^* 


Date  Due 

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